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مراجعه ماث تالته اعدادي ترم اول 2027 | revision on unit 1 math prep 3 first term | math prep 3 — Transcript

by Eng. Mostafa Math - مهندس الماث · 22,069 words · 2,816 segments · language en · Watch on YouTube

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0:00The full semester package price is 525
0:04EGP.
0:05>> Fine, fine, I won't say anything. Fine,
0:06you continue with them, I won't spoil
0:07the surprise. Alright, I'll continue
0:10with them. Peace, mercy, and blessings
0:12of God be upon you. Prep 3 students,
0:14Engineer Al-Maqath students, how are
0:15you doing? I hope you are well. This
0:17video, guys, is not like any other
0:18video. Of course, it's something great,
0:20as I've brought you a review of the
0:21first unit with explanation and
0:22solutions, but that's not the surprise.
0:24Reviews come and go, but the big
0:25surprise is that we've reached 100,000
0:27on YouTube. Okay, enough clapping and
0:29cheering. Enough, enough. Honestly, I
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1:01price is 525 EGP. And if you don't know
1:03the platform, search for "Mr. Mustafa
1:04Hossam platform" on Google. And the
1:06matter is very easy. So this will be
1:07for the YouTube students who follow me.
1:09As for my dear ones who are already
1:10subscribed with me, they won't be able
1:12to benefit from this offer. Because
1:13this offer is for those who haven't
1:14subscribed yet. But I have made another
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1:17forget them. It is a 50%discount on the
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1:21This will be available on 15/10 and
1:2316/10. So you will contact me on the
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1:28details. Guys, let's focus. Those who
1:30haven't subscribed with me before will
1:31have this offer. But if someone is
1:33already subscribed and has paid, they
1:34have another offer. It is a 50%discount
1:36on the final review course. And this is
1:38to be contacted via WhatsApp. As for
1:40those who are not subscribed and want
1:41this offer, you will find it on the
1:42platform. Let's get back to our topic.
1:44That we have a review for the first
1:46unit. A very beautiful unit consisting
1:47of two lessons. I've made a perfect
1:49review for it. A review that includes
1:51all the questions related to the unit.
1:53So you will benefit a lot, God willing.
1:54And the most important thing is not to
1:55forget to hit like, subscribe, and turn
1:57on the bell button. So that whenever a
1:58new review or a new video is posted,
1:59you get a notification. And so that the
2:01video reaches the largest number of
2:02people and everyone benefits. Let's
2:04start, in the name of God, the Most
2:05Gracious, the Most Merciful. We will
2:07start our review, guys, with some laws.
2:11Some things we took last year, and some
2:13things related to this year. It doesn't
2:15matter. What matters to me is that I
2:17know these laws very well. Because it
2:19will help me in my problems for unit 1,
2:22okay. Okay, so I have here, for example
2:24, 2 power 3 multiplied by 2 power 4.
2:27Basically, guys, if I have a number,
2:30let's say 2 and it has a power of 5, we
2:33agree that what's on top is called what
2:36? Power. Listen, the one at the bottom
2:39is called the base, so if I say the
2:41word "base," you understand that the
2:43base is the one at the bottom. And the
2:45power is the one at the top. Anyway,
2:46the first rule is very, very easy. Very
2:48easy. It tells you that if you have the
2:50base equal to the base, and they are
2:52multiplied. The answer remains the same
2:55base. And you add the powers. 3 plus 4
2:58gives me 7, beautiful stuff. Let's see
3:02one just like it. But if they are
3:03divided, and the base is the same as
3:05the base. The answer is the same base.
3:07If they are divided, then you subtract
3:09the powers. So, 5 minus 2 gives me 3.
3:12Now, if I have 2 multiplied by 3, and
3:14all of that is to the power of 4,
3:16whatever the power is, it distributes
3:19here and here, so the answer is 2 to
3:21the power of 4 multiplied by 3 to the
3:23power of 4, it's that simple. Honestly,
3:27guys, I want to stop here for a moment
3:29because there is a point that some
3:32people get confused about. So I must, I
3:34mean, academic integrity requires that
3:36I make you pay attention to it. Which
3:38is, what are the two examples? One: x
3:42multiplied by 3, all to the power of 2.
3:48And I'll also give you x plus 3, all to
3:51the power of 2. There is a huge
3:54difference, guys, between these two.
3:57When there is multiplication between
3:59them, whatever the power is, it goes
4:00down and distributes here and here. So
4:03the answer is x squared multiplied by 3
4:06squared, done. But when inside the
4:09bracket that has a power of 2, there
4:11are several terms. Meaning this is
4:13called a term, and this is called a
4:14term. What differentiates one term from
4:17the next is having a plus or minus sign
4:19here. Nice. So in this case, this is a
4:23rule we took before, which is expanding
4:25a bracket to the power of 2. It's a
4:28rule, so what do you do? The first to
4:30the power of 2, and the second to the
4:32power of 2, and put it at the end. And
4:34always, always put a plus before it.
4:37The sign here comes down as it is.
4:39Whether it is a plus or a minus. If it
4:40is a plus, it comes down as a plus. If
4:42it is a minus, it comes down as a minus
4:43. But this is what? Always a plus. Good
4:45. And in the middle, we do 2 multiplied
4:48by the first, multiplied by the second.
4:51So you will do 2 multiplied by x
4:53multiplied by 3, so 2 multiplied by 3
4:55is 6, and the x comes down. So I should
4:58put 6x here. Guys, this is not in Unit
5:011, it is not there. But this is
5:03something you already took before. You
5:04took it in the first year of middle
5:06school. And it was repeated for you in
5:07the second year of middle school. A
5:08bracket to the power of 2 is something
5:10very important for you to know how to
5:11expand. So there is a difference
5:12between when I give you a bracket with
5:14two terms inside it to the power of 2.
5:16Or when there is only one term inside.
5:18What does one term mean? It means there
5:19is no plus or minus between them. No,
5:21rather, there is multiplication or
5:22division. In that case, the power just
5:24distributes normally. This is what we
5:25have in the unit, that the power
5:27distributes normally. This is not for
5:29us. But it is something old, so you
5:30should be aware of it. It is likely
5:32that you will encounter it in the rest
5:33of the curriculum. Alright, guys. So I
5:35just want to draw your attention.
5:38I’ll group similar things together
5:39for you so you don’t get confused.
5:41Okay. For example, if I have 2 over 3,
5:43all to the power of 3, it’s very
5:44straightforward. Just like I
5:46distributed the power of 4 here.
5:49Because they were being multiplied. So,
5:50the power of 4 went onto the 2 and the
5:523; if they are being divided, it gets
5:53distributed normally too. Here, it is
5:55considered division. But it is just
5:57written in fraction form. So, the 3
5:59goes onto the 2 and onto the 3,
6:00becoming 2 to the power of 3 over 3 to
6:03the power of 3; it's that simple. Of
6:05course, these things have further steps
6:06to solve. But that is not the point.
6:08I’m just showing you how the story of
6:10powers works. Now, look at this. This
6:13is the new one for this year. The first
6:15lesson: Power of a Power. What is it?
6:17If you have two powers stacked on top
6:19of each other. Meaning, if you have 5
6:21squared, all to the power of 4, these
6:24two powers are multiplied. So the
6:26answer for something like this is 5.
6:28All right. I’ll multiply the 2 by the
6:314, giving me 8, and that’s how I
6:32expanded it. This is called power of a
6:34power. It’s actually the first lesson
6:35in your curriculum. Okay. Here, for
6:37example, you have 5a squared multiplied
6:39by 2 cubed, and all of this is raised
6:41to a power; the 2 here is considered a
6:43power of a power. Meaning a power that
6:45is supposed to have powers under it. No
6:47problem. For me, the power of 2 goes
6:50here and goes here. Just like I was
6:53doing here, guys. So, I have 5a squared
6:55, and I’ll apply another power of 2
6:58to it. Multiplied by 2 cubed, and
7:02I’ll apply the 2 to it. Great. So the
7:06answer is that you have two powers here
7:08stacked up. 2 times 2 gives me 4. So
7:11it’s 5a to the power of 4 multiplied
7:14by 3 times 2 gives me 6, meaning 2 to
7:16the power of 6; the whole point is to
7:19show you how to handle powers. Okay.
7:23This is also considered similar to the
7:24previous one. But instead of them being
7:26multiplied, they are now being divided.
7:29No problem at all. Here is the division
7:31. The power of 3 will go here and here.
7:34Shall we do it in one step to see if
7:36you understood? I have 2 to the power
7:38of 5. And it will have a power of 3
7:41above it, meaning it will be 2 to the
7:43power of 5, all to the power of 3, so
7:45the 5 and 3 are multiplied. So it
7:48becomes 15 over here. 3 squared, and it
7:52will have a power of 3, so you have 3
7:55as the base. With a power of 2 above it
7:58, and a power of 3 above that, so the 2
8:01and 3 are multiplied, becoming 6, very
8:03simply. Now, there was a rule, guys,
8:05that says what? If you have a, Look
8:10here. If you have something with a
8:13negative power. We always want to get
8:16the simplest form, guys. When you have
8:18something with a negative power, I
8:20can't call it the simplest form. So I
8:21have to do something about it. How do I
8:22handle it? You use this rule. What does
8:25this rule say? It says that if, for
8:27example, you have 2 to the power of -3,
8:29what do I do? I'll make it 1 over 2 to
8:32the power of 3. What happened? I put 1
8:35on top, and took all of this and put it
8:37underneath, but I removed the negative
8:39from the power. This is a rule: if you
8:42have a base with a negative power, make
8:45it 1 over that base. And the power,
8:47instead of being negative, becomes
8:49positive. Instead of the power being
8:51negative, it became positive, and I
8:53placed it over the value. So this is
8:55the rule: if you have a to the power of
8:57negative n, it is 1 over a to the power
9:00of n, okay? One more thing: if you have
9:022 over 3, which is a fraction, and it's
9:04to the power of a negative number. You
9:06want to get rid of this negative, so
9:08you flip what's inside, and it becomes
9:103 over 2, and the negative power
9:12becomes a positive power. The last rule
9:14I can talk about, and then we'll solve
9:16some problems. There are still rules in
9:18the unit, of course, but I've given you
9:20a few to solve some problems, then I'll
9:22give you other rules to solve more, so
9:24you'll find I've divided everything
9:25very nicely. Okay, I'm telling you that
9:28any number to the power of 0 equals 1;
9:30this is a rule to keep in mind. 5 to
9:33the power of 0 is 1, negative 3 to the
9:36power of 0 is 1. That's it. Okay, so
9:39I've brought you a question here. This
9:40is a question to watch out for; it
9:41comes in exams and is very important.
9:44It says 5 to the power of x minus 3
9:46equals 1. I want to know what x is. To
9:49be honest, you could take a calculator
9:52and type 5, with a power above it in
9:54place of x. Try any value. I'll try 2,
9:56for example. 2 minus 3 gives me 1 over
9:595. Let's try 3 instead of 2. For
10:02example, it gave me 1. Oh, so x is 3,
10:04but of course, this method is random
10:06and won't always work for you. However,
10:09I'm showing you that you can manage.
10:11But the correct way, which we will use,
10:14is this: If you want to find x, focus.
10:18I have 5 with a power, and the answer
10:21comes out as 1. Well, I know that any
10:24number to the power of 0 gives 1, so I
10:27know that x...Minus 3 is what? Zero. It
10:31became a very simple equation. Move the
10:33negative 3 to the other side. So x
10:35equals what, my friend? 3, so pay close
10:38attention to this problem. When it
10:40gives you a number as a base with a
10:42power. This power has an unknown like x
10:44, for example. Their answer is 1; this
10:46means that the whole power is equal to
10:49zero. Okay, it's asking for another
10:51requirement. It says, come on, find me
10:522 to the power of x. But I already
10:54found x to be 3. So it wants 2 to the
10:57power of 3, which, when you calculate
10:59it on the calculator, gives me 8. Let's
11:02look at another problem. It's the same
11:05idea. You have 7 to the power of x
11:07minus 2 equals 1. Yes, any number to
11:10the power of 0 gives 1, so I know that
11:13x minus 2 is zero. Move the 2 to the
11:15other side. So x equals 2. What does it
11:18want? I want 5 power x, which I found
11:21to be 2, and then 2 minus 3. So the
11:23answer is, calculate it on the
11:25calculator right away. It will give you
11:281 over 5, so the answer is 1 over 5.
11:32That's it. Alright, let's look at the
11:34next one. It says, which of the
11:37following is equal to this. Solve, I
11:39mean solve. You have a to the power of
11:422, then to the power of 4. When there
11:44are two powers one on top of the other,
11:45what do you do? Multiply, well done. So
11:47it becomes a ^ 2 to the power of 4,
11:49which is 8, I multiply them together.
11:52Multiply by a power 5, well, there's. A
11:55rule that also says if it's the same
11:57base and they are multiplied. Then the
11:59answer is the same base. Since they are
12:02multiplied, we add the powers. When you
12:04add 8 and 5, it gives you what? 12,
12:09that's it guys. So the answer is a
12:12power...wait, what did I say? I said 12
12:15, no, the answer is 13, sorry. Okay. So
12:19the answer is a power 13, are we done
12:22guys? Okay. Let's see the next one.
12:27Here. It tells you I want the result of
12:29this expression. One step at a time.
12:31Basically guys, when I have a bracket
12:33like this. A bracket here with a power
12:36of 3 and this bracket has a negative, I
12:38must. Think, why? I think because the
12:41behavior of the power makes the
12:43negative do something specific. In what
12:46sense? Well, if the power is an even
12:49number, it eliminates the negative. If
12:53the power is an odd number, it keeps
12:56the negative as is, or moves it outside
12:59. How does it move it outside? I'll
13:01tell you, for me, this 3 is odd, so
13:03what does it do to the negative? It
13:05moves it outside. I write it completely
13:08outside like this and write the
13:09fraction, that's the way. Now, one step
13:12at a time, I have the 3, it should be
13:14distributed to the 2, so it becomes 2
13:17power 3, the 3 lands on the a power 2,
13:19wait, I have a, the a. a has a power of
13:222 and a power of 3 lands on it, so they
13:24are 2 and 3 stacked. So I will multiply
13:27them, so it becomes 6, okay. And below
13:30I have 2 multiplied by a, which is 2a.
13:33Fine, it's normal now. You see how you
13:35want to solve it. The matter is simple.
13:37You have 2 power 3, wait. Just a moment
13:40. Look, look. Since you have the items
13:42present on top and bottom. Both the 2
13:44and the 2 and the a. I will start now
13:47by canceling the fraction. And I will
13:49work with division. This, when you have
13:522 power 3 divided by 2 power 1, it
13:54becomes 2 power. When I perform
13:56division between them. I subtract the
13:58powers. So 3 minus 1, and the same
14:00story with the a, here it is. And I
14:03will do 6 minus 1. So it remains
14:07negative 2 power 2, a power 5. All I
14:13will do now is calculate 2 power 2,
14:15which is 4, and I have a power 5 here,
14:17so I have solved my problem, that's it
14:19guys. He wanted the simplest form. So I
14:22got him the simplest form in this way.
14:25Pay attention, if this power 3 were an
14:26even number, say for example it was 2,
14:28what would happen to the? This negative
14:30would completely disappear from the
14:31problem. But because it was an odd
14:34number, it pushed that negative outside
14:36like this, okay guys? Alright, let's
14:38look at the next one. You have x power
14:432, power 3, and you have power 4, power
14:470. Basically, before I even think, it's
14:50power 0, everyone. Are you following?
14:53Power zero, and anything to the power
14:55of zero is 1, and I brought the plus
14:57sign down. Now, here you have x power 2
15:00, power 3; the 2 and the 3 are
15:01multiplied, so that's power what? Power
15:046, so this is the final answer, which
15:06is this. Okay. Let's see the next one.
15:11It's a very simple topic, actually. The
15:12first unit is very easy, guys. We still
15:15have many more questions. Okay, it says
15:18what is the value of n power 2, m power
15:213, all to the power of 3; the 3 is
15:23distributed here and here, so it
15:26becomes n. Right. The n already has a
15:29power of 2. And I added a power of 3 to
15:32it, so it becomes 2 and 3, which are
15:34multiplied to be 6, and here is m. It
15:38already had a power of 3, and I added a
15:41power of 3, so there are two powers
15:44stacked, 3 and 3, which are multiplied
15:46to be 9, good. After finding that, it's
15:50supposed to want the whole value. It
15:52says it wants the value of this
15:53expression. Not just for you to adjust
15:55the powers. Well, sir, do I have the
15:57values of n and m? You do. It says n
16:00equals m equals 2, so I will remove n
16:03and put 2, so it becomes 2 power 6, and
16:05I remove m and put 2, so it becomes 2
16:08power 9, multiplied together. Is that
16:11good? Show the steps, don't just
16:12calculate it directly on the calculator
16:14. So the answer is 2, and you add the
16:17powers, 6 plus 9 gives me 15; now get
16:20your calculator and compute 2 power 15.
16:24You'll find it gives you a somewhat
16:26large number. No problem, that's what
16:29it wants. If you want to calculate this
16:31too, just to make sure you're on the
16:32right track. Here is 2 power 6
16:34multiplied by 2 power 9, it gave me the
16:36same number. So I'm on the right track,
16:38thank God. Meaning I did 6 plus 9 and
16:40got 15, so I'm doing it right. Let's
16:43look at the next one. It says what is
16:46the value of...which of the following
16:48is equal to this expression? This is
16:51very easy. You have b with a power of 2
16:53, then a power of 3, two powers stacked
16:56so they are multiplied. So the answer
16:59is power 6, the answer is power 6, very
17:01simple. Now here it says this
17:06expression when x is 1, so step by step
17:09. First, x has power 11, and power 6, I
17:13will multiply 11 by 6, which gives 66;
17:17it says when x is 1, put x as 1, so 1
17:21power 66, so the final answer is 1,
17:24calculate 1 power 66, 1 to any power is
17:281, finished guys. Keep in mind that all
17:31these questions I'm solving with you
17:32are part of the general review. You'll
17:34find them as questions from assessments
17:36, the textbook, things from me, or from
17:38the El-Moasser book. You'll find that
17:39I've gathered everything for you, so
17:41you'll be very happy. Okay, here it
17:44says simplify. Alright, I have Y power
17:478 multiply. You have here Y to the
17:49power of 3, all to the power of 4.
17:52These are multiplied, so it becomes Y
17:54power 12 because 3 times 4 is 12, the
17:57answer will be Y. And I will add the
17:59powers, so it's Y power 20, that's the
18:02final answer. That concludes this
18:05problem. Alright, stay with me here. I
18:08have a negative here that is inside a
18:11bracket. This bracket has an even power
18:14, so the negative will vanish, vanish
18:17guys. And the power of 2 will descend
18:19here and here. You have M to the power
18:22of 3, and I applied the power of 2 to
18:24it. So 3 and 2 are multiplied, making
18:27it 6, and I will apply the power of 2
18:30to the N. Problem finished. Okay, all
18:33of this is considered an application of
18:35the rules we've learned. Now, let's
18:37start getting into slightly bigger
18:39things, but they are still easy. You
18:41have here X power 4 to the power of 5,
18:45so we multiply these two powers, 4 and
18:485 become 20. Step by step, this is a
18:51bracket. To the power of 4, the
18:54negative vanishes, guys. Why? Because 4
18:56is an even power, so it eliminates the
18:59negative. Great, so the negative will
19:01vanish. And the 4 will apply to the X.
19:03Meaning you have X to the power of 3,
19:04and you raised it to the power of 4.
19:06Two powers on top of each other are
19:08multiplied, so it becomes 12, so the
19:11answer is X power 20, X power 12, so
19:13the answer is X, and since there is a
19:15divide, I subtract the powers. 20 minus
19:1812 gives me 8. That's my first question
19:22. The next question, pay attention now.
19:25I want to tell you something honestly.
19:26There is a difference, guys, before I
19:28solve this problem. There is a
19:30difference when I give you, for example
19:32, negative A to the power of 6, or if I
19:34give you negative A power 6, there is a
19:36difference between these. And be
19:39careful, we said, sir, that an even
19:41power gets rid of the negative. Yes,
19:43that's when it is applied to it. What
19:45does it mean "applied to it"? Meaning
19:47here, the 6 is a power for whom? It's a
19:50power for the A and for the negative.
19:53Because it's in a bracket, so it's a
19:55power for everything inside the bracket
19:57. So it is a power for A and for the
20:00negative, so the power can control the
20:02negative as it pleases. So if it is
20:05even, it will eliminate the negative.
20:06Therefore, the answer is A power 6,
20:08nice words. But look at this one, I
20:11have negative A to the power of 6. The
20:16power 6, guys, is only on the A; it has
20:18nothing to do with the negative. Is the
20:20power 6 on the negative? No, sir, it
20:22isn't. Do you want to say it's there?
20:23Not at all. This is what was on the
20:25negative, which is in a bracket, and
20:27the 6 is on the bracket. So, the
20:28negative is affected by the six because
20:31there are brackets. But here, everyone,
20:34the six is only on the a. Meaning, this
20:36is its answer, right? You keep the
20:38negative as it is, and the a power 6
20:40remains a power 6, so it's considered
20:42ready as is. Pay attention to this
20:44point because it is very important.
20:46Okay, for example, here I have negative
20:48a power 5; it is negative a power 5 as
20:51is. What am I going to do with it?
20:53Right, multiply. Here there are
20:56brackets, and the brackets are raised
20:57to the power of two, which is an even
20:59power, right? This means the two is a
21:01power on the a power 6 and a power on
21:04the negative. So when it is an even
21:06power, what does it do? It gets rid of
21:07the negative. And when it goes onto the
21:10a power 6, a is to the power of 6 and
21:11then 2, so it becomes 12 because I
21:13multiply them together. All right, let
21:16me erase this. Well, this negative is
21:21outside, I have nothing to do with it.
21:23I have a and a, so the answer will be a
21:25. Since they are being multiplied, I
21:27add the powers 5 plus 12. Now I have
21:31solved my problem, guys. Okay, let's
21:35look at the next question. Here, I have
21:40b's on top and bottom. With power of a
21:43power. I will work step by step. I have
21:45the first one b squared to the power of
21:483, so it becomes b power 6, multiply by
21:51the b 3. And it's raised to 4, so I do
21:533 times 4. So it is b power 12. And the
21:57b 5 raised to 2 becomes 10, that's just
21:59to start, to finish the power of a
22:02power. Now I have b's on top and bottom
22:04. You can follow these steps. Either
22:07you work on the top ones together, then
22:09the top with the bottom. Or you do it
22:11all at once; I will tell you both. Look
22:14, I have b power 6 multiplied by b
22:16power 12, so the answer is b, and you
22:19add the powers 6 plus 12. And at the
22:23bottom, I have b power 10. This will
22:26equal b, and I subtract, which gives me
22:28what? It gives me this method. There is
22:32another method, let me erase. What is
22:35the other method that gets it done
22:36quickly? I have b, b, and b, so the
22:40answer is b, and what do I do? Any
22:43power on top, put a plus in front of it
22:45. Any power on the bottom, put a minus
22:47in front of it. Look here, I have 6, I
22:49will write a plus before it. Of course,
22:51the first plus is not important, but
22:53okay, let's organize things. 6 is a
22:55power on top, so I put a plus before it
22:57. 12 is a power on top, so I put a plus
23:00before it. The 10 is a power on the
23:03bottom, so I put a minus before it,
23:05here is the minus. Then you use the
23:08calculator to do 6 plus 12 minus 10 by
23:10itself on the calculator, it gives me 8
23:13, so I know the answer is b power 8. So
23:15, the answer, everyone, will be b power
23:178. This can be applied to any problem
23:20as long as you have the same variable
23:22on top and bottom. Even if I had, for
23:25example, a multiplied by this, let's
23:27say a power 6, I finished the problem,
23:29but I am giving you extra info to
23:31organize things for you and so you
23:33understand how I think. A to the power
23:36of 7, and I have, for example, A to the
23:38power of 5 multiplied here, can I apply
23:40this, sir? Yes, absolutely. You take
23:43the Bs and solve them. What about the
23:46As? Go ahead and write them as usual.
23:48Here is the A. I have the power 7 on
23:50top, so I put a plus before it. And the
23:52power 5 is below, so I put a minus
23:54before it. The answer is A. 7 minus 5
23:57gives me 2, so I basically made you
24:01another problem. Just so you understand
24:04that this rule can be applied normally.
24:05As long as the term exists both above
24:07and below. Even if there are other
24:08things next to it, it is perfectly fine
24:10. Well, let's look at the next question
24:12. It says we want to find the simplest
24:15form. Let's take it one step at a time.
24:18Basically, guys, this 3 is distributed
24:21to everything inside. Meaning it goes
24:24here, here, here, here, and here. Let's
24:28perform this operation now. I have 2 to
24:32the power of 3 applied to it. And I
24:35have A to the power of 2, and I brought
24:37power 3 on top of it; 2 times 3 gives
24:39me 6, okay. And I have B to the power
24:43of 2, and I brought power 3 on top of
24:44it; 2 times 3 gives me 6, lovely. Over,
24:48I have B to the power of negative 1,
24:50hold on a moment. Negative 1, I brought
24:54power 3 on top of it; negative 1 times
24:563 gives me negative 3. And here is,
25:00sorry, here is A with power 4, I
25:03brought power 3 on top of it, so 4
25:05times 3 gives me 12; same story I told
25:08you earlier. This is what I will work
25:11with. I have 2 to the power of 3, which
25:13is the numerator. 2 to the power of 3
25:15gives me 8, so it is written here. Be
25:16careful, if it were the denominator, I
25:18would write it differently. Meaning if,
25:19for example, the 2 to the power of 3
25:21were here below, stay with me. If it
25:24were here, I wouldn't just write it on
25:26the line as 2 to the power of 3, no
25:28honey, or even write 8, no baby, I
25:29would write 1 over 8 because it is the
25:32denominator. But honestly, my problem
25:35has it so that...The 2 to the power of
25:373 is on top, which is fine, okay. So
25:39for me, when I calculated 2 to the
25:41power of 3, it gave me 8, I wrote it on
25:42top like that, no need to write over
25:44anything anymore. Now, I have A
25:45existing on top and bottom. And I have
25:47B existing on top and bottom. Write one
25:50A and write one B. Here are the As. Any
25:55power on top, put a plus before it. So
25:57this 6 becomes positive 6, and any
25:59power below, put a minus before it.
26:02Meaning the 12 becomes minus 12. As for
26:05the Bs, this power number is on top. So
26:08I put a plus before it, positive 6.
26:11This is negative 3 below. Be careful
26:14because some people get confused in
26:15this part. I told you, any power below,
26:17put a minus before it. The value of the
26:19number doesn't concern me. So what do I
26:21tell you? I tell you the number below
26:24is negative 3, so is that finished? No.
26:26The number below is negative 3 and you
26:29put a minus before it. So it becomes
26:32minus minus 3, exactly like that. So, I
26:34will write it for you again: minus
26:37negative 3. It was negative 3 in the
26:39bottom, I wrote it and put a minus
26:42before it. There we go. But simply,
26:45start doing your calculations. You will
26:48do 6, calculate it on the calculator: 6
26:51minus 12 is negative 6, minus negative
26:543 is 9. Are we done? No. It wants the
26:58simplest form. And I agreed with you at
27:01the start of the review that the
27:02simplest form cannot have a negative
27:05power. So this is not the simplest form
27:07. You must deal with it. Wait, I have 8
27:09, there is no problem with it. And I
27:11have b to the power of 9, no problem
27:13there. This a to the power of negative
27:156, I will write it as 1 over a to the
27:17power of 6; these things are multiplied
27:19together, guys. Alright, have we
27:22finished now, honestly? There is one
27:24last step. Anything written on the line
27:27next to a fraction moves to the top as
27:29long as they are multiplied, guys.
27:31Meaning, honestly, the final answer. I
27:34should move the 8 up here. And move b
27:37to the power of 9 up. So it becomes 8 b
27:40to the power of 9 over a to the power
27:43of 6. This is the simplest. Final form.
27:48Pay attention, I will explain a point
27:50to you. If you have, for example, 5.
27:53And I asked you, is this just 5? I
27:55asked you, is this a numerator or a
27:58denominator? You would say a numerator,
28:00why? Any number in the world standing
28:03like that is considered standing on the
28:06line, so it is considered a numerator
28:08because it's like over 1, guys, like
28:11over 1. So if I have 5 multiplied by 1
28:13over 2, honestly, this 5 should go up
28:16here next to the 1, so it becomes 1 by
28:185 over 2. The 1 doesn't matter. It's 1
28:21by 5 is 5 over 2. That's why when I
28:25moved the 8 and the b, I ignored the 1.
28:28It's 8 and multiplied by b, so they are
28:318 and b to the power of 9. So this is
28:33the final answer, guys. Anything
28:35written on the line. Move it to the top
28:37for me. As long as it's next to a
28:39fraction, of course, they must be
28:41multiplied. So no one is foolish enough
28:44to have here 8 plus, for example, 1
28:46over a. Does this have a solution? No,
28:48this is as far as it goes. Are you
28:50going to tell me to move it up? No, my
28:51son. They must be multiplied for you to
28:53be able to move them to the top. But if
28:55it's like that, then no, this is one
28:56term and that is one term. Focus with
28:58me on what I'm saying. I am clarifying
28:59everything for you. Keep in mind, I am
29:01actually building your foundation.
29:02Meaning, I am reviewing for you and
29:03building your foundation at the same
29:04time. Because these small things I'm
29:06saying aren't in the curriculum. These
29:08are supposed to be old things you
29:09already know. But I am drawing your
29:11attention to them. Because they grow
29:13with you. You start entering the coming
29:15years, or even this year. You will be
29:16exposed to things like this, and you
29:17need to understand them. Watch out.
29:20Alright, let's see what's next. A few
29:25easy multiple-choice questions,
29:27hopefully. But they depend on a law.
29:30And I'm practically done now. The first
29:33part about the laws will surely be
29:34needed in the rest of our problems.
29:36However, there is a new part I want to
29:39talk to you about. It relates to the
29:42second lesson in the unit. You know,
29:43this unit has two lessons. The part I'm
29:45going to explain relates to the second
29:46lesson. What is it? Look, my dear,
29:48focus with me. I'll explain it to you
29:50right here now. Guys, if I have, for
29:53example, x to the power of 2 equals 4.
29:58Power 2. Wait, let me change that to
30:02power 3, sorry. I have x to the power
30:07of 3 here and 4 to the power of 3 here,
30:09and I asked you, what do you think x
30:11could be? Very intuitively, you'd tell
30:13me, "Sir, since this is like that, then
30:15surely this is like that." So, how much
30:17is x? It's 4. A very logical answer and
30:19your statement is 100%correct. From
30:21here, we conclude a rule: if the power
30:23is equal to the power, then the base is
30:25equal to the base. Another rule, if I
30:28have 2 to the power of x equals 2 to
30:31the power of 5 and I asked you, what do
30:34you think x is here? You'd tell me, "
30:36Sir, it's very intuitive." Since this
30:38is like that, then this is like that.
30:40So x is 5, correct. And this is the
30:42second rule. Or let's call this the
30:44first one because I have them ordered
30:45in my head a certain way and I want you
30:46to memorize them in this same order.
30:48This is the first one, which is what?
30:49It says that if the base is equal to
30:51the base, then the power is equal to
30:53the power. What does the second one say
30:55? If the power is equal to the power,
30:57then the base is equal to the base.
31:00Does that sound good? These are the two
31:02rules. There is a third one I want to
31:04tell you about. Focus with me on it,
31:05and don't forget it because it always
31:06comes up in the exam. Which is what?
31:08Look, let me give you an example of it.
31:10If I have 5 to the power of x equals 7
31:13to the power of 0. Stay with me for a
31:17moment. No, I'm just adjusting it. Here
31:19it will be x. Step by step. Okay, let's
31:25start. 5 to the power of x, and 7 to
31:30the power of x in this form. I should,
31:35as soon as my eye lands, on something,
31:36it lands on the things that are similar
31:37. I have the power is equal to the
31:39power. So naturally, like rule number
31:41two. Be careful with this. One, the
31:43first rule, the second rule, the third
31:46rule. Naturally, as soon as I see the
31:49power is equal to the power, I say the
31:51base is equal to the base. So I look at
31:55the base: is 5 equal to 7? Is there
31:57such a thing? No. For example, here,
32:00when I said the power is equal to the
32:02power, the base is equal to the base,
32:05so x is 4. Beautiful, no problem. But
32:08here, guys, and I actually want to find
32:10x. Okay, walk with me here, I will say
32:13the power is like...the power, so the
32:15base is like the base. But can I say
32:18that 5 equals 7? No, I can't. In this
32:23case, the third rule appears, which
32:26says if you find that the power is
32:28equal to the power, and the base cannot
32:30be equal to the base as happened here,
32:33then you take the power and equate it
32:35to zero. So x equals zero. There are
32:38problems for this that we will solve
32:39when we reach them. These are the three
32:41rules, guys. We must pay close
32:43attention to them because there are
32:45many questions on them. Well, let's
32:46look at the first question on them. I
32:48don't want to over-explain, let's just
32:50solve some problems instead. The first
32:53question gives us 2 power x equals 8.
32:55If I ask you, what do you think x is?
32:58You’ll tell me, "Honestly, Mister, I
32:59could do one trick." I'll take my
33:01calculator, type 2 power, and test it
33:03out. 2 power 2 gives me 4, 2 power 3
33:06gives me 8, so x is 3, problem solved.
33:10Okay, if it’s a multiple-choice
33:12question, that works. But it won't
33:14always work, because sometimes the
33:16problem gets more complex than 8, maybe
33:18with other things added on. Regardless,
33:20that isn't the only way, nor is it the
33:22correct way. That method is fine if
33:24you're stuck and just want to
33:25experiment. But what is the right way?
33:29Listen up, because there is a "Golden
33:31Rule" I use a lot in all the upcoming
33:33questions for the rest of the unit.
33:35Guys, what is this Golden Rule? What
33:37does it say, Mister? It says that when
33:40you find that the powers are unknown,
33:43make the numbers smaller. Again, when
33:47you find the powers—here I have an x,
33:49an unknown—you make the numbers
33:51smaller. What does "make the numbers
33:54smaller" mean? It means that,
33:56truthfully, you should turn 8 into 2
33:58power 3. I shrunk it; instead of it
34:01being 8, a big number, I made it 2
34:03power 3, which I’ve memorized. And I
34:06have 2 power x here. So now, the first
34:10rule: if the base is equal to the base,
34:12then the power is equal to the power.
34:14So what is x? It is 3. Listen to the
34:19Golden Rule because it is very
34:20important. This is Mister Mustafa’s
34:21Golden Rule, the Math Engineer’s rule
34:23; nobody else will tell you this. When
34:25you find the powers are unknown, make
34:28the numbers smaller. We made the
34:30numbers smaller; we shrunk the 8, and
34:32the 2 was already small. I’ll pause
34:34for a moment to talk about this "making
34:36numbers smaller" part. Mister, how do I
34:38know if this number is big enough to
34:40make it smaller? Look, my dear, the
34:42small numbers for me in this unit's
34:44problems are 2, 3, 5, and 7—the first
34:47four prime numbers. These are the small
34:50numbers for me. So, I look at 8, for
34:52instance, and I want to shrink it into
34:54one of these. I turned it into 2 power
34:563. But Mister, I haven't memorized them
34:58! You, Mister Mustafa, know that 8 is 2
35:01power 3, you know 16 is 2 power 4, you
35:03know a bunch of things. I don't know
35:05anything. Well, there is a method for
35:06that. There is a way for me to take a
35:08number and make you shrink it. How? Let
35:10me show you. Take the 8 and do
35:13something called factorization. What
35:15does that consist of? How do I break it
35:18down? Look, my dear, 8 is an even
35:21number, so choose 2 for it. You choose
35:24from those numbers, so choose 2. When I
35:27do 8 divided by 2, it gives me 4.
35:29Repeat the process once more. 4 is an
35:32even number, so choose 2 for it. When I
35:36do 4 divided by 2, it gives me 2. Great
35:38, 2 is an even number, so choose 2 for
35:41it. When I do 2 divided by 2, it gives
35:44me 1. What happened? Guys, I broke the
35:478 down into three 2s. So what do I do?
35:51I write the 2, and see how many times
35:54it repeats: 1, 2, 3, so it’s power 3.
35:56That’s how I broke down the 8 to be
35:59power 3. This method works for any
36:01number, really, any number. Okay,
36:05excuse me, sir, why did you always
36:07choose the 2? Why not choose the 3, 5,
36:09or 7? No, each one has a specific use,
36:12which is what? Look, here we had 8,
36:14that was my starting number, what kind
36:17of number is it? Even, so I choose an
36:19even number for it. What if it were odd
36:21, sir? Then I choose an odd one for it,
36:22which is the 3. But what about...5 is
36:24odd too. No, listen to me for one
36:26second. If it's even, choose 2. If it's
36:30odd, choose 3, good. If it ends in 5 or
36:340, choose 5. What does that mean? If I
36:37give you 50 and say we want to factor
36:40it. You look at this digit. What is
36:43this, my dear? It's 0, whether it's 0
36:47or 5, I factor using 5. So I put 5 here
36:50. Then you do 50 divided by 5, which
36:53gives me 10. I look here again. This is
36:570, I choose 5 for it. 10 divided by 5
37:02gives me 2, good. This 2 is even, I
37:06choose an even number for it, which is
37:082. Divided, it gives me 1, nice and
37:11easy. The 50, guys, I gave you an
37:13example for it. I broke the 50 down
37:16into 5 and 5. Meaning I write 5 once.
37:18And see how many times it repeats, one,
37:20two. So it's power 2 multiplied by...
37:22There's one more part I must include,
37:24which is who? The 2. So the 50 was
37:27broken down into small numbers: 5 power
37:302 and also 2. Right, we will take many
37:33things and I will factor them for you.
37:36But I'm just drawing your attention to
37:37something. How to know how to pick the
37:39first number here. I have an even
37:41number, I pick an even one for it. What
37:43if I had 9, for example? This is an odd
37:45number, I pick an odd one for it, which
37:47is the 3. Okay, so we've agreed. If I
37:51have an even number, I pick 2. If I
37:53have an odd number. I pick 3. If it
37:56ends in 0 or 5, one of the two, I pick
37:595. When, sir, do I pick 7 if 3 doesn't
38:02work? Meaning, if I tried 3, guys, and
38:06it didn't work. Like where? Like if I
38:09get 49, okay. I look here and find the
38:139 is odd. Good. 49 is odd, the 9 is odd
38:16. So I pick 3. You try to calculate 49
38:20divided by 3, it won't give me an
38:22integer to put here. It won't work. So
38:26that means 3 failed with 49, so then
38:29you'll have to take the 7. There is the
38:347, okay. And you do 49 divided by...7,
38:36it gives me 7. Let me repeat the
38:38process again. The 7 is odd, so I take
38:40the 3 with it. Okay, I do 7 divided by
38:433 and find it doesn't work. Yeah, so
38:46honestly the 3 doesn't work. So I
38:48resort to 7, 49 divided by 7 gives me 1
38:51, so you see 49 is broken down into 7
38:54repeated twice. The repetition is
38:57placed on top, 7 repeated twice.
38:59Alright guys, that's the topic of
39:00simplifying numbers in a nutshell. And
39:02the answer to this question, as we said
39:04, is 3. Let's take more questions. For
39:07example, I have here 4 power x equals 8
39:10; you could try the options. But this
39:12question might not necessarily be
39:13multiple-choice. It could come as an
39:14essay question. So if it comes as an
39:16essay, I must be ready to show steps.
39:18Okay, don't you ever forget the golden
39:20rule. I swear, if you forget the golden
39:23rule of Unit 1, which is: if powers are
39:26unknown, simplify the numbers. If you
39:28forget it, you're done for. Staring
39:30blankly at the question. Not knowing
39:31what to do. I'd be lost like that.
39:33Consider this question has no options.
39:354 power x equals 8, what do I do?
39:37Should I guess? You won't be able to
39:39guess. There are no options, after all.
39:41The question is an essay type. So what
39:42do I do? What do I do? The golden rule
39:44is what will sort everything out for
39:46you. Which is what? If powers are
39:48unknown, simplify the numbers. So, do I
39:51have the 4 here, is it considered small
39:53or large? Large. I told you, what are
39:55the small numbers? Only four, there are
39:57no others. The 2, the 3, the 5, and the
40:017. These are the ones if I see them, I
40:03won't simplify them. That's it, they
40:04are. These are small for me. But the 4,
40:06for instance, no, it needs to be
40:07simplified. Let's simplify it then, the
40:104. It's an even number. I'll take the
40:12even factor for it, which is the 2. 4
40:14divided by 2 gives me 2, the 2 is an
40:15even number. I'll take the 2 for it. 2
40:18divided by 2 gives me 1, so the 4 is
40:20broken down. Into 2 repeated twice.
40:24Okay, so the 4, guys, I will make it 2
40:27power 2, and it has a power x, sir, no
40:29problem there. Okay. Equals 8, we just
40:33did it a moment ago, 2 power 3. Alright
40:37, I have two powers on top of each
40:38other, be careful how you handle them.
40:40When there are two powers on top of
40:41each other, I multiply them. 2 by x, 2x
40:46. And here, 2 power 3, the base is like
40:50the base, so the power...Just like the
40:52power rule, guys. If the base is like
40:54the base, then the power is like the
40:56power. So 2x equals 3, you move the 2
40:59to the other side. It's stuck there by
41:01multiplication, so it goes over by
41:03division. So x equals 3 over 2. That's
41:07it. x equals 3 over 2, easy guys, easy.
41:12Easy questions. Of course, with my
41:15great explanation, you'll find
41:16everything easy. The next question,
41:18guys, it gives me 3 power negative x
41:20equals 3. Honestly, guys, this question
41:22has a simple idea. It doesn't differ
41:24much from the previous one. But it does
41:26have a simple idea. So I'll ask you to
41:28let us do a very nice trick. And
41:29honestly, I've done this in all the
41:31videos uploaded to YouTube. And you
41:32guys on YouTube are happy with it.
41:34That's why I always do it for you.
41:36Which is what? I leave a question for
41:38you to solve. And I’ll go into the
41:40comments and pick someone. From those
41:42who solved this question correctly to
41:43win the term for free. We’ve done
41:45this, by the way, guys, in all the
41:46previous videos I posted. We had
41:48winners and announced them on Facebook.
41:50And we replied to them. I told them you
41:51won, so get in touch with us. And we
41:53activated the entire term course for
41:55them for free. Okay, I’d like to ask
41:58you to pause honestly. Pause and solve
42:01it, then write in the comments. Then
42:02watch as I solve it now. Okay, in the
42:04name of God, this is a very easy
42:06question. You have 3 to the power of
42:08negative x equals 3, so the base is the
42:11same as the base, meaning the power
42:12equals the power. You’ll tell me, "
42:15Mr., there’s no power here." My son,
42:17no; if a thing doesn't have a power, it
42:18means it's to the power of 1, that's it
42:20. Okay, so I will make negative x equal
42:241; base matches base, so power matches
42:27power. Negative x equals 1; look at
42:31this negative if you get confused here.
42:33Some people get confused in this part.
42:35Guys, this is like negative 1, okay? So
42:38this negative 1 is stuck to the x. It
42:40goes to the other side. Since it's
42:41attached by multiplication, it goes
42:43over. By division, so it becomes So x
42:48equals negative 1; that’s it, so what
42:51is x, man? x is negative 1; keep in
42:53mind this is just me explaining, not a
42:55big deal. Some other people, for
42:58example, see negative x equals 1 and do
43:01what? They already understand that I am
43:05multiplying both sides. When I do this,
43:08it goes away, and when I multiply by
43:11negative 1, the negative appears. So
43:14it’s like you took that negative and
43:16put it over there. You put it there
43:18without changing the sign or anything.
43:19I just mean the basis is that it's a
43:20negative 1 attached by multiplication
43:22that went over by division. That is the
43:24basis. Some people just memorize it
43:25straight away like that: if they see a
43:26negative, they move it. Straight away,
43:27just as it is, the negative. That’s
43:29also a trick, so okay. Just keep in
43:31mind these simple little things. Okay,
43:34let’s see the next one. You have here
43:374 to the power of 2x equals 1 over 8.
43:39Well, I’ve memorized my golden rule.
43:42When you find unknown powers, make the
43:44numbers smaller. I want the unknown
43:46power, which is this x, so I must make
43:48the numbers smaller for myself. The 4
43:50will become 2 squared, and it already
43:52has a power, which is 2x. And I have
43:56the 8; it must be made smaller, it is 1
43:57over 8. We made it smaller; it will be
44:002 cubed. Honestly, guys, but initially,
44:04I have 2 and 2x, these must be
44:06multiplied. Here is the base, and I
44:09will multiply 2 by 2x, 2 by 2 is 4, so
44:114x. Well, if I wrote this as 1 over 2
44:16cubed, I haven't benefited at all.
44:18Normally, what do I want to say? I want
44:21to say the base equals the base, so the
44:22power equals the power. But I have this
44:25here on the line, while this is a
44:26denominator. So, honestly, I want to
44:28manage. Manage to do what? You could
44:30take that 2 cubed and bring it up, then
44:32change the sign of the power. Don't we
44:34know, guys, that if I have 2 to the
44:36power of negative 3, it is the same as
44:381 over 2 to the power of 3? Yes, I'm
44:41used to the fact that when I find a
44:43negative power, I convert it to 1 over
44:45the term without the negative sign; I
44:47change its sign, I change its sign.
44:49Okay, what if it comes to me like this?
44:51Can I change it back to suit the
44:53problem's needs? I mean, is it allowed?
44:56So here is 2 to the power of 4x equals,
44:57watch what I'm going to do now. This
44:59will become 2 to the power of negative
45:003. Did I change anything? No, I just
45:03adjusted the form to fit my problem. So
45:06I have the base matching the base,
45:09which means the power must equal the
45:11power. So I will say that 4x equals
45:14negative 3. There it is, 4x equals
45:16negative 3. I'll move the 4 to the
45:18other side by division, so it becomes
45:21negative 3 over 4, which is this. Are
45:24we good, guys? Alright, let's look at
45:28another problem, with the same concepts
45:30. Pay attention, we are solving a
45:32problem here. For any question, show
45:34your steps, man; no guessing or
45:36anything like that. Okay, let's take it
45:38one step at a time. Looking here, I
45:40have something on the left and
45:41something on the right. The natural
45:43golden rule tells me if the powers are
45:46unknown, simplify the numbers. And I
45:48already know this type of problem. I
45:50will simplify the numbers, and if the
45:52base matches the base, then the power
45:54equals the power. Well, for me, the 2
45:57is small, but the 16 needs to be
45:58simplified. Let's do it this way; here
46:01is 16, what am I simplifying it to?
46:03Remember with me: 2, 3, 5, and 7. Great
46:07, for the 16, I always look at the
46:09number on the right. The 6 is even, so
46:11I take the even factor, which is 2.
46:13There's nothing else, it's the only
46:14even one among them. 16 divided by 2
46:17gives me 8; the 8 is even, I take 2, it
46:19gives me 4. 8 divided by 2 gives me 4;
46:21the 4 is even, I take 2, 4 divided by 2
46:24gives me 2; the 2 is even, I take 2, 2
46:26divided by 2 gives me 1; so the 16,
46:29guys, has been broken down into 2s.
46:31Here is 2, repeated how many times?
46:33Four times, 1...So it is 2 to the power
46:37of 4. Now let's write it here; I have 2
46:40to the power of x + 2 equals 16; I'll
46:42put 2 to the power of 4 instead, with a
46:45bracket 3 minus x. Okay. Basically, you
46:50have two powers here on top of each
46:52other, they must be multiplied. Watch
46:54out now. These aren't just two numbers
46:56that I'll multiply and be done with. No
46:57, the 4 is going to be multiplied by
46:59this whole thing. Honestly, if I went
47:02and did it on the side, it would be...
47:07In this format. We've learned before
47:10that when I have something stuck to a
47:11bracket, and inside there are two terms
47:13. Meaning this is a term and this is a
47:15term. So this 4 goes in here and it
47:17goes in here. 4 by 3 is 12, 4 by
47:21negative x is negative 4x. So you
47:25should, when you multiply this by this,
47:27get this answer. I did it for you on
47:29the side. I did it for you on the side.
47:31So I have 2 and its power. When I
47:35multiplied these, it gave me 12 minus
47:384x. And I have here 2 to the power of x
47:42squared, very nice. Thank God, I
47:44reached the point where the base equals
47:46the base. So, the power equals the
47:49power. Great. So, x squared equals 12
47:53minus 4x. Let me pause for a moment. We
47:57want to bring back memories from the
47:59second year of middle school. We
48:03learned, guys, when I have an equation
48:06with x squared, x, and a number. At
48:10that point, I must do something called
48:12factorization. Okay, remember with me,
48:14I’m reminding you now. Factorization.
48:16If you remember this factorization, you
48:17would get, for example, an x squared, x
48:19, number problem. I’ll tell you like
48:20what. Like, for instance, x squared
48:23minus 5x plus 6 equals 0. When
48:25something like this comes to you, you
48:27can't tell me, "Sir, I'll solve it by
48:29taking the 6 to the other side with a
48:31negative sign." It won’t be solved.
48:34Because you’ll have x squared minus
48:365x equals negative 6, and then these
48:39have no solution; they are stuck like
48:41that. So, that wasn’t the way to
48:43solve it. Rather, the correct way to
48:45solve it was to do factorization. How?
48:48Factorization used to give you two
48:50brackets, two brackets here, equal to
48:53zero. So, you would put x here and x
48:55here. And honestly, to...get the
48:57numbers that go here. There were many
48:58ways. But in the third year of middle
49:00school, we use the calculator. These
49:01are old things. I don’t need to waste
49:03time on them in third-year middle
49:05school problems. So, how do I get them
49:06from the calculator? Focus with me. Of
49:08course, I said it in the classes. But,
49:10you know, we are reviewing and I am
49:11saying everything. You have something
49:14called "Mode" on the calculator next to
49:16the "On" button. Press it. Then, choose
49:18the word "EQN." Which is either number
49:205 or number 3. Depending on your
49:22calculator. For me, it is number 5. I
49:24pressed it. Four things appeared. I
49:26will choose number 3. So I did Mode 5 3
49:29; some others might do Mode...And the
49:33word "EQN" is at number 3, so they do
49:36Mode 3 3. I have Mode 5 3, so three
49:39boxes appear like this. You are
49:45supposed to enter into these three
49:46boxes...the numbers in order like this.
49:48The one next to x squared is 1, so you
49:51write 1 here, so I write 1 and press
49:53equal. The one next to x is negative 5.
49:55So I put negative 5 and press equal.
49:58The number by itself here is 6, so I
50:00put 6 and press equal. It tells you x1
50:02is 3. I am giving you an example
50:04outside, okay? x is 3 and x is 2. These
50:10are the answers, by the way. These are
50:11the answers. Aren't you trying to solve
50:13this equation anyway? To get the x. Am
50:15I right or wrong? Right. So, it’s the
50:18calculator. Once you enter 1, negative
50:205, and 6 on Mode 5 3, it outputs the
50:23answers. But, of course, to do the
50:25factorization steps. I have 3, I flip
50:27its sign and put it here, so it becomes
50:29negative 3. I have 2, I flip its sign
50:31and put it here, so it becomes negative
50:332. I split it, once x minus 3. is zero.
50:35And once, x minus 2 is zero. So you
50:38move the negative 3 over as positive.
50:39And you get that X is 3. So you move
50:41the negative 2 as a positive. And it
50:43turns out that X is 2, and this is
50:45factorization. You had many methods for
50:48factorization. You took them last year.
50:51So, for factorization in Unit 1
50:53specifically, I won't have a problem,
50:55God willing. Because it's a simple
50:57problem. Unit 2 is full of
50:59factorization, full of it. So that's
51:01fine, no problem. We now know what
51:03concerns us in our unit. The important
51:05thing is when you get something like
51:06this. As soon as you see X squared and
51:09X in it, you have to resort to
51:11factorization. The order of the
51:13factorization problem must be X squared
51:16, X, number. X squared, X, number,
51:20equals 0; that's the order. So,
51:23honestly, I must take the negative 4X
51:24and move it inside here. So I get X
51:27squared plus 4X. And the 12, I move it
51:31inside too, minus 12 equals 0. You can
51:34solve this, it's very easy. Mode 5 3,
51:37and the three fields appear. Enter 1
51:40and press equal. 4 and press equal.
51:42Negative 12 and press equal. The
51:45calculator showed me that X1 is 2 and
51:47X2 is negative 6, these are the answers
51:50. But since this is an exam problem, I
51:53must show steps. Two brackets equal
51:56zero. I put X and X. Here is 2, I
51:58always change its sign. Whatever is
52:00here, I change its sign and put it in
52:01the bracket. Here is negative 6, change
52:04its sign and put it here. You split it:
52:06once X minus 2 equals 0, so X is 2,
52:08which is what I already got from the
52:10calculator. And once X plus 6 equals 0,
52:13so move the positive 6, and X comes out
52:16as negative 6. Now I have the value of
52:19X. He asked to find the solution set.
52:22Excuse me guys, let me erase this extra
52:24part. It was a problem from my own head
52:25. After finding X is 2 and X is
52:28negative 6, you just have to write them
52:31inside your solution set. Which is 2
52:34and negative 6. He wants the solution
52:36set, meaning get me the X values. We
52:39got the X values, thank God. A very
52:40important problem, guys. Let's pay
52:42attention to it. No, no, I can't catch
52:45my breath. Okay, let's continue, let's
52:47continue. But I'll do two things. First
52:49thing, I will drink some water. I want
52:51to tell you guys that I am actually
52:53sick. I have a cold, so...At the same
53:05time, in the reviews, I'm interested in
53:07telling you everything. But quickly so
53:10the review doesn't get too long. And
53:13you don't miss anything either. You
53:14watch it as if you were attending the
53:16explanation. From the beginning of the
53:18year, don't worry at all. So, we said
53:19we'd do two things: drink water. And
53:26the second thing is, I want to liven
53:28things up a bit; let's refresh
53:30ourselves. I brought you a game today
53:32you'll like a lot. I'm not sure if you
53:34played it when you were little or not?
53:36But hopefully, you'll know it. I used
53:38to love it very much when I was little.
53:42I love it very much. Maybe some guys
53:44now, forget about the beginning because
53:45it's just...It starts differently, but
53:47as soon as you open it, you’ll know
53:49it. Young people might be playing it
53:51now, but the new version, I’ll tell
53:52you what game it is. I don’t want to
53:54say what game it is. Never mind. Look
53:56at what’s written, isn’t that the
53:57game name? May God make it easy. This
54:02sound, if you used to play it, you’ll
54:04recognize it. This sound, alright,
54:06start game then. New game. Guys, this
54:13game...
54:13>> Buying it.
54:14>> Oh, wow.
54:17>> GTA Vice City, I used to play it and I
54:19love it so much. Guys, we want to skip.
54:25We want to skip this chatter. Let’s
54:28go, stay with me now. Honestly guys, I
54:32really hope the sound is okay. I adore
54:36this game. And mind you, I love Vice
54:39City because there was San Andreas,
54:41then there’s GTA 5 which is usually
54:44on PlayStation, and 6 just came out.
54:47Let’s see what we’re going to do.
54:49We have a car, let’s hop in, man,
54:51take your time. I like to keep driving,
54:55but driving here is very difficult. Oh
54:57my God, look at the gas and brakes here
55:00. Wait for me a second. Whoa. I’m
55:08telling you, I swear it’s very hard.
55:10I want someone now, I want someone to
55:12play with. I swear. I hope it stays.
55:19The missions don't matter. Yeah, they
55:21don't matter, it's not the most
55:22important thing. No, seriously, it’s
55:24hard. I’ll just get out and walk,
55:26it's better. I can't drive. Running is
55:31easier then. How long are we going to
55:37keep walking? We need a car. Wait, look
55:39, they even prepared the cheats for you
55:41. Let’s read, "heavies" are the
55:46weapons, "full armor" is the shield. I
55:50got armor now, okay. Wait, wait, "raise
55:54wanted level," what’s this? That’s
55:58when the police are after you. That. I
56:02made it, look, look here, I made it
56:04four stars. Let’s run, because the
56:09police are going to finish us off.
56:12There are planes and everything. I used
56:17to love, whoa, no, no guys, easy now,
56:19don’t do that. How should I handle
56:23this? I can’t, guys. They’re on me,
56:26they’re swarming me, I swear, what a
56:29mess. I surrender, they’re on me,
56:32okay, I surrender. If I get out,
56:41they’ll crush me. The car is
56:45overturned anyway, they can’t open
56:47the door. Wait a minute. Are we "busted
56:57," guys? It’s not just "busted,"
57:00that’s enough. I’m tired, my nerves
57:03are shot, my nerves are shot, man, I
57:05swear.
57:06>> I’m telling you, isn’t this the
57:08police, that’s great. I want to see
57:12people inside. He’s telling me, what
57:15is this? He’s shooting at me,
57:19actually, by the way, whoa.
57:25>> Enough, guys, shoot back because you
57:27are just...
57:30>> Look, they’re headbutting, guys. Just
57:35get this gun. Pistol. Stop it. Take a
57:43break now. Take a break, I’m tired of
57:45you. He’s shooting in general. I want
58:01to reload, I think. I’m telling you
58:04what? My nerves are shot. That's it,
58:05that's enough. I swear my nerves are
58:07fried. What is this? What's going on
58:09here? I just wanted to chill out a bit.
58:12I wanted to chill out, but my nerves
58:14are shot too because I'm not in control
58:15. I really love this game. But since
58:18I'm not in control, I'm going crazy.
58:20Alright, let's calm down. Calm down,
58:22calm down. Honestly, I want to play
58:23something light right now. But not now,
58:25maybe a little later we'll play
58:26something else. Let's get back to our
58:28problems. Oh God, I feel like I was
58:30dreaming, guys. You don't understand.
58:32When I stand here, the screen makes me
58:34feel like I'm inside the game itself.
58:36The screen fills, it fills my whole
58:39face. So I, uh, I can't see. You see
58:42from far away, so it's fine for you.
58:44There is no power but in God. Why do I
58:46do this to myself? I swear all this was
58:47just to entertain you, man. I hope you
58:49don't get entertained. I hope you don't
58:50get entertained. I, my nerves are shot.
58:53Also, I shouldn't have let the whole
58:55police force chase me. Okay. Anyway,
58:58let's get back to our problems. I hope.
59:00I feel like. I've scrambled your brains
59:02. You won't remember anything from the
59:04explanation now. You won't remember
59:05anything from the explanation. I hope
59:06it's for the best. God willing. Let's
59:09start, in the name of God, the Most
59:10Gracious, the Most Merciful. Come on,
59:12focus with me. Forget what happened.
59:14Wipe it from your mind. Look, my dear.
59:17We currently have 5 to the power of x
59:19squared minus 4. Alright, and. 125 to
59:23the power of x. The golden rule is if
59:27the powers are unknown, shrink the
59:29numbers. Okay, wait a moment for me.
59:32Here is 5, and here is x squared minus
59:354. It's small. There's no problem with
59:37it. Now, the 125, we want to shrink it.
59:39Let's do it together. Here is the 125.
59:44I look here, I agreed with you that if
59:47the number is 0 or 5, I choose 5 for it
59:51. 125. Divided by 5 gives me what, guys
59:56? It gives me 25. No, it gives me 25.
1:00:05Nice. I look here, if this is 5, I
1:00:08choose 5. Divided by 5 gives me 5. This
1:00:11is 5, I choose 5. 5 divided by 5 gives
1:00:15me 1, so the 125. 125 broke down into 5
1:00:18repeated three times. 5 to the power of
1:00:203, how beautiful. So here is 5 to the
1:00:22power of 3, with power x on it. By the
1:00:25way, this is similar to the previous
1:00:26problem. You must have forgotten the
1:00:27previous problem because of the chaos
1:00:29that happened. The previous problem was
1:00:31the same idea. That you have here 5 to
1:00:33the power of x squared minus 4 equals
1:00:36these multiply each other, so 3x. The
1:00:40base is like the base. So the power is
1:00:43like the power. You just need, since
1:00:46this is a problem with x squared, x,
1:00:47and a number. To adjust its order. So
1:00:50that it's x squared then x. So I must
1:00:53move this over here, guys. So I have x
1:00:55squared. This is positive 3x, move it
1:00:57as negative 3x. And you have negative
1:00:59four. The equation equals 0, so do mode
1:01:025 3 to find your x-values. And that way
1:01:04, you have obtained the solution set.
1:01:07Mode 5 3, here we go. Mode 5 3, I'll
1:01:10enter 1, negative 3, and negative 4. It
1:01:14tells me that x is four. And that x is
1:01:18negative 1, here are two brackets. x
1:01:23and x. Here it is 4, so it's negative 4
1:01:26. Here it is negative 1. So it is
1:01:28positive 1, write these down as steps.
1:01:30Look, so what is this part here? Before
1:01:33x equals 4, all of this should be steps
1:01:35. Move the negative 4, so x is 4. You
1:01:38have x plus 1 equals 0 here, move the 1
1:01:41to the other side, so x is what?
1:01:43Negative 1. Alright everyone, we have
1:01:47solved it. But don't you forget. Pay
1:01:49attention now. Pay attention, this
1:01:51question will appear on the exam. Great
1:01:52, thank God, it's an easy and nice
1:01:54question, sir. You solve the question
1:01:57and get the x's, and you forget, you
1:01:59forget to get the solution set. You
1:02:01forget to get the solution set. You
1:02:04forget to get the solution set. Never
1:02:07forget to get the solution set. Never
1:02:12forget to get the solution set when it
1:02:13is asked of you. Never, I don't know
1:02:15what else to do for you. Okay. So what
1:02:19are we going to do? Solution set, we
1:02:21put inside it 4 and negative 1, guys it
1:02:24is mandatory. Why, my dear friend? You
1:02:27solved the whole question. A question
1:02:29worth three marks, for example. Then
1:02:31you lose a mark because of the solution
1:02:32set. Why, isn't that a shame? It is a
1:02:34shame. Alright, thank God. Now, what
1:02:36was that horror that happened? Now, we
1:02:39won't forget the solution set, God
1:02:40willing. Let's see the next question. A
1:02:45clear and direct question. It says I
1:02:47have 7 with an exponent, and 7 with an
1:02:49exponent. The base is equal to the base
1:02:51, so the exponent is equal to the
1:02:53exponent. It's a very easy question, by
1:02:55the way. This exponent is equal to that
1:02:57exponent, because if the base equals
1:02:59the base, the exponent equals the
1:03:01exponent. I have 2x plus 3 equals x
1:03:04minus 4, the x's are scattered here and
1:03:07there, that doesn't work for me. I must
1:03:11get the x's on one side and the numbers
1:03:13on the other. So I will bring this x,
1:03:15it is positive x here, it goes as
1:03:16negative x. Look, I have 2x, I brought
1:03:19negative x to it. And the positive 3
1:03:22goes there as negative 3, so I have
1:03:24negative 4 minus 3, 2x minus x, two
1:03:27oranges minus one orange leaves one
1:03:30orange. Negative 4 minus 3, calculate
1:03:33it on the calculator and it's negative
1:03:357, we found that x is negative 7, if
1:03:37you want to verify your answer, by the
1:03:39way, and in previous questions too,
1:03:41after finding x for example, calculate
1:03:43it here. So I will do 7 to the power of
1:03:452 multiplied by negative 7 plus 3, it
1:03:47gave me a number like this, it doesn't
1:03:49matter, it's strange but it's okay, it
1:03:52is 5.0 times. 10 to the power of
1:03:53negative 10, when I calculate the other
1:03:55side where x is negative 7, it should
1:03:57give the same number. Look at that. 7
1:04:00to the power of negative 7 minus 4,
1:04:03yeah, it gave me the same number. So, I
1:04:06solved it correctly. Because the
1:04:08negative seven I found made the left
1:04:09side equal to the right side. What I
1:04:11just did was a confirmation process,
1:04:13nothing more, guys. But the problem is
1:04:14already finished. Well, there are some
1:04:18other ideas one needs to solve with you
1:04:20so you can pay attention to them. Here,
1:04:23I need to find the value of m. Well,
1:04:27the golden rule remains the only way.
1:04:29If you have the power as an unknown,
1:04:32make the numbers smaller. For me, this
1:04:34one is small. You have 3 to the power
1:04:36of negative 2m plus 1, this is fine.
1:04:39Multiply, the 9 can be made smaller.
1:04:42Shrink it, you already know the methods
1:04:43. It will be 3 to the power of 2,
1:04:46generally 3 squared to the power of 2m,
1:04:50alright. And the 27 below will also be
1:04:52made smaller. It will be 3 to the power
1:04:55of 3. Let me show you the 27. Here is
1:04:58the 27, it's odd, so choose 3 for it.
1:05:01Divided by 3 is 9, the 9 is odd, choose
1:05:033 for it. 9 divided by 3 is 3, the 3 is
1:05:06odd, choose 3 for it. 3 divided by 3 is
1:05:091. The 3 is repeated three times, so
1:05:12it's 3 cubed. So, I broke 27 down to 3
1:05:16cubed, good so far. But I want to
1:05:18multiply them together, so it becomes 3
1:05:21squared to the power of m. Anyway guys,
1:05:25let's tidy up what we can. For me, I'll
1:05:29move this; the 1 over 3 cubed won't
1:05:32help me. Instead, I need to bring the 3
1:05:35cubed up. So it becomes 3 to the power
1:05:38of negative 3. You are missing one
1:05:42simple thing. You have the bases. What
1:05:44are these bases? The bases are the same
1:05:47. There is multiplication between them.
1:05:48You add the powers. So here it is. And
1:05:52do negative 2m plus 1 negative...2m
1:05:54plus the first power plus the second
1:05:56power. Which is 4m equals 3 to the
1:05:59power of negative 3. Okay. Now you just
1:06:04need to calculate negative 2m plus 4m,
1:06:06which is 2m. If you want to calculate
1:06:08it now, no problem. If you want to
1:06:10calculate it when equating power to
1:06:11power. It's fine, it won't matter.
1:06:12Let's calculate it now. This will be 3,
1:06:15look guys, negative 2m and 4m are like
1:06:18terms. So their result is also m, and
1:06:20negative 2 plus 4 is 2. Plus 1 equals 3
1:06:23to the power of negative 3. The base
1:06:26equals the base, so the power equals
1:06:29the power. Meaning 2m plus 1 equals
1:06:32negative 3. Solve it, teacher. Move
1:06:35this to the other side with a negative
1:06:37sign. So negative 3 negative 1, so 2m
1:06:39equals negative 4. You will move the 2
1:06:41to the other side by division. So m
1:06:44equals negative 4 divided by 2, which
1:06:46is negative 2. How beautiful. Do we
1:06:49want to verify, sir? Verify, my dear.
1:06:51Take the m as negative 2 and substitute
1:06:53it here. So look, I'm calculating now.
1:06:553 to the power of negative 2 by
1:06:57negative 2, since m is negative 2, plus
1:06:591, step down from the power. Multiply 9
1:07:03power 2, multiply m by negative, what,
1:07:05negative 2. I calculated that, it came
1:07:08out to 1 over 27. Allahu Akbar. My
1:07:11solution is 100%correct. And that's
1:07:14perfectly fine. That's perfectly done.
1:07:17Alright, the next question. Was this
1:07:19question asking for the solution set or
1:07:20was it asking for m? It was asking for
1:07:22m. And this question was asking for the
1:07:24solution set. After I found x, I put it
1:07:26in a solution set. That's it. I just
1:07:28want to get through this. I mean, I
1:07:29don't want to keep wasting time on
1:07:30things I've repeated. Because there
1:07:32were many questions above that asked
1:07:33for the solution set. I even kept
1:07:34telling you, don't forget the solution
1:07:36set. So I just want to finish. The most
1:07:39important thing is that you don't
1:07:40forget the solution set in the exam. We
1:07:41want to move quickly now. We are on
1:07:42this question. Yeah, let's go to the
1:07:45next one. Look here, it says, solve the
1:07:49equation in front of you. Well, to
1:07:51begin with, I need to simplify the
1:07:53numbers. There are things I already
1:07:54know now. 8 will be 2 power 3
1:07:56multiplied by 3 power x minus 2 equals
1:07:5927, which is 3 power 3. Mister, you
1:08:01know these things because you memorized
1:08:04them? Yes. But you, afterwards, or in
1:08:07these problems and so on. You're going
1:08:09to have to simplify them. You take the
1:08:1127 and simplify it. We've done it many
1:08:13times. So I won't keep repeating it.
1:08:16Okay, 27 is 3 power 3 multiplied by 2
1:08:18power x minus 2. That's it, my friend.
1:08:22What do I do for a question like this?
1:08:24You'll tell me, "Mister, the base
1:08:25equals the base, so the power equals
1:08:26the power." That's 100%correct. So what
1:08:30do I do, boss? I have 2 power 3 here,
1:08:33and here 2 power x minus 2. So the base
1:08:36equals the base, so the power equals
1:08:37the power. Meaning x minus 2 is 3, and
1:08:41look here, I have 3. Power x minus 2
1:08:45equals 3 power 3, so x minus 2 equals
1:08:483. So fine, it's considered the same
1:08:50equation. Solve it then. Move this to
1:08:52the other side, so x is how much, boss?
1:08:54It's 5. And this will also give you
1:08:56that x is 5. I've solved my question,
1:08:59and thank God. I just want to
1:09:02double-check. I found that x is 5. I
1:09:03want to calculate here: 8 multiplied by
1:09:063 power 5 minus 2 gives me this. When I
1:09:08calculated for x equals 5, it gave me
1:09:10this. And this is 27 multiplied by 2
1:09:15power, we said x is 5 minus 2. The same
1:09:20. Thank God, I solved it correctly.
1:09:22Let's look at the question. It asks for
1:09:24the solution set here. So write it down
1:09:26, put 5 inside it. Okay, it says if 2
1:09:31power x minus 3 equals 1, people forget
1:09:34and come and say, "Mister, I want to
1:09:36see the base equal to the base." So the
1:09:40power equals the power. They forget
1:09:42that you already have a rule. What does
1:09:44it say? It says that if you have 2.
1:09:47Power 0, the answer is 1. So as soon as
1:09:50I see the answer 1, I know the power
1:09:52here is 0, meaning x minus 3 equals 0,
1:09:55so x is 3. Question finished in seconds
1:09:58. Okay, guys. Let's look at the next
1:10:01one. A simple question, let's read it
1:10:04together. It says, a rectangle with
1:10:06dimensions. A rectangle is something
1:10:08that has a length and a width. Right.
1:10:11It says the dimensions are 2 to the
1:10:12power of x + 1 and 11; let's say this
1:10:14is the length and this is the width. Or
1:10:17this is the length and this is the
1:10:18width, it makes no difference. And it
1:10:19says the area, my friend, is 176. x.
1:10:24Okay, I know the formula for the area
1:10:27of a rectangle is length times width. I
1:10:29have the area, which is 176. And I have
1:10:35the length, which is 2 to the power of
1:10:36x, multiplied by the width, which is
1:10:3811. It has become a normal equation;
1:10:40you want this, so I'll move it to the
1:10:42other side with division. So, 2 to the
1:10:45power of x. I brought 176 down.
1:10:48Calculation. It gives me 16. So, here
1:10:56is 2 to the power of x + 1, guys, we've
1:10:59done this. Before, 2 to the power of 4,
1:11:01since the base equals the base, the
1:11:03exponent equals the exponent. Meaning x
1:11:04+ 1 must equal 4. You move the positive
1:11:071 to the other side as negative. 4
1:11:09minus 1, so x is 3. We solved our
1:11:12problem, and thank God. You can go back
1:11:15to the original equation to verify. Go
1:11:16ahead, here is 2 to the power of 3.
1:11:19Plus 1, multiplied by 11, gives me 176.
1:11:23So we solved it correctly, thank God.
1:11:27Now, the two questions in front of you
1:11:29talk about rule number three. When I
1:11:33told you about it, what did I say?
1:11:35Normally, if the exponent equals the
1:11:36exponent, then the base equals the base
1:11:38. This is rule number two. What does
1:11:41rule number three tell you? If the
1:11:43exponent equals the exponent and I
1:11:45cannot say the base equals the base,
1:11:47then I take the exponent and set it to
1:11:49zero. Let's see. I see here, guys, that
1:11:53the exponent equals the exponent. So
1:11:57naturally, I should say the base equals
1:11:58the base. But here the base is 7 and
1:12:00the base is 5; can I say they are equal
1:12:02? Of course not. So you know this is
1:12:05rule number three. And I will take the
1:12:07exponent and set it to zero. Move this
1:12:09to the other side. So x equals negative
1:12:111. I solved my problem. Same story here
1:12:14. You have the exponent equal to the
1:12:16exponent. Naturally, we would say the
1:12:19base equals the base. But I can't say 5
1:12:22equals 3, so we take the exponent and
1:12:24set it to zero. So x equals 1. Easy,
1:12:29guys, that's it. Let's look at the next
1:12:32one. We will start entering a type of
1:12:35question, guys. Which is in front of
1:12:38you looking like this. For example, it
1:12:39comes and asks you for the simplest
1:12:40form. Something like this. And apply my
1:12:43golden rule. When you find the
1:12:45exponents are unknown, simplify. The
1:12:47numbers, don't forget this golden rule.
1:12:49Because it is very important. You find
1:12:50the exponents are unknown, simplify the
1:12:52numbers. So for me, I simplified 4
1:12:55before. It was 2 to the power of 2. And
1:12:58it already has an exponent of x, okay.
1:13:00And I also simplified 9 before. It was
1:13:023 to the power of 2. And it has 2 minus
1:13:05x on it. We haven't simplified 6 before
1:13:08. So let's try to simplify it, come on.
1:13:11Here, 6 is an even number. I choose
1:13:14from 2, 3, 5, and 7. Since 6 is even,
1:13:17I'll choose an even number for it,
1:13:18which is 2. 6 divided by 2 gives me 3;
1:13:213 is odd, I'll choose an odd number,
1:13:24which is 3. 3 divided by 3 gives me 1;
1:13:26that’s it, it's done. What is this?
1:13:29Well, 6 didn't result in a repeated
1:13:31number. So I would say it's something
1:13:33to a power. That repetition didn't
1:13:35happen, but 6 was broken down. Into 2
1:13:37and 3. So, 2 multiply 3, is that normal
1:13:40? Yes, it is normal. What else would I
1:13:42do? I take whatever comes out here. If
1:13:45I get. Different things, they are
1:13:47multiplied together; for example, and I
1:13:49mentioned this in the first review,
1:13:52look, if I got 2 and 2, this part here,
1:13:54then it would be 2 repeated twice. One,
1:13:57two. So it would have a power and
1:13:59multiply normally. There is no problem
1:14:02at all. Alright, the important thing,
1:14:04guys, is that we know 6. Will be broken
1:14:06down into 2 by 3, here you go, 2. By 3,
1:14:10and here is power 2x. What will happen
1:14:13now? This 2x should be distributed here
1:14:17and here. Is that clear, so it becomes
1:14:202? 2 by x is 2x, multiply 3 to the
1:14:25power of 2 minus x. Look, I could tell
1:14:29you, let's take it aside and multiply
1:14:31them together. You distribute this here
1:14:34and distribute this here. But I want
1:14:35you to be faster than that. You already
1:14:37know that 2 is multiplied by this whole
1:14:39expression. So you multiply the 2 here
1:14:42and here as it is, instead of taking it
1:14:44out and adding unnecessary work to the
1:14:46paper. So here it is, it's correct, no
1:14:49problem. But I want you to be faster
1:14:50and more efficient. 2 by 2 is 4, 2 by
1:14:54negative x is negative 2x. Is that
1:14:58clear? Over, I have a 2 here that will
1:15:00get the 2x power. So 2 power 2x,
1:15:03multiply 3, which will also get the 2x
1:15:06power. Now look, be sharp, nice,
1:15:09handsome, sweet, smart, cool, and
1:15:10relaxed. The 2 power 2x cancels out
1:15:13with the 2 power 2x. Pay attention,
1:15:17what am I going to tell you? I will
1:15:18tell you to pay attention. Alright, you
1:15:20have 3 and 3, so the answer is 3, the
1:15:22power above minus the power below. So
1:15:25the power above is 4 minus 2x. That's
1:15:28the top one, minus the power below,
1:15:31which is 2x. Guys, I want to tell you
1:15:33something. When you have a power above
1:15:36minus a power below, if the power below
1:15:39consists of several terms, honestly, I
1:15:41should have opened a bracket. For
1:15:45example, let's assume the power here
1:15:47was x plus 1, so when I put the minus,
1:15:49I shouldn't write 2x plus 1 directly.
1:15:51No, I must open a bracket. So again, if
1:15:54you have a minus to be placed, and
1:15:56several terms will follow it, like
1:15:58these one, two terms here. You must
1:16:01open a bracket because the negative
1:16:02goes inside here and goes inside here
1:16:04to change the signs. But if it's just
1:16:06one term, then it's fine, you don't
1:16:08need a bracket. Okay, so what is the
1:16:10answer to this question? Look, my dear,
1:16:14you now have 3 power 4, and you have
1:16:17negative 2x, negative 2x. These are
1:16:21like terms, so their result is x.
1:16:23Calculate negative 2 minus 2 on the
1:16:25calculator, you get negative 4, and
1:16:27that’s it, I’ve solved it and got
1:16:29the final answer. It asks for one more
1:16:32thing, it says you want the result when
1:16:35x is 1, okay, so 3 power 4 minus 4
1:16:37multiplied by 1, calculate it on the
1:16:39calculator, and the power will result
1:16:41in zero. So what is the final answer? 1
1:16:44. Type all of this on the black
1:16:46calculator. You will find it gives you
1:16:481, and that’s how we solved our
1:16:50question. Let's look at another one
1:16:52like it, because the upcoming questions
1:16:54are of the same type. Okay, step by
1:16:58step. For the 9, of course, my golden
1:17:00rule is to want the simplest form. When
1:17:02you find the powers are unknown, make
1:17:05the numbers smaller. The golden rule,
1:17:08as they say, helps you solve most of
1:17:09the questions in the unit, God willing.
1:17:13Whether it’s that the base equals the
1:17:14base, so the power equals the power, or
1:17:16the three rules we were just using
1:17:17earlier, or problems like this. Alright
1:17:20guys, so the golden rule is very
1:17:22important. 9 becomes 3 power 2, and it
1:17:26has the power 2n minus 1, multiply the
1:17:2918, we want to make it smaller. Let's
1:17:32make it smaller on the side. Here is 18
1:17:34, it's even, so we choose 2, 18 divided
1:17:38by 2 is 9, it's odd, so we choose 3, 9
1:17:41divided by 3 is 3, and we choose 3, 3.
1:17:45Divided by 3 is 1, so the 18, guys, has
1:17:47been factored into a 2. And a 3
1:17:50repeated twice. 3 repeated twice with a
1:17:53multiply between them. By the way, if
1:17:55you want to be sure, calculate this on
1:17:56the calculator. See if it gives you 18
1:17:58or not. So you can also be sure that
1:17:59you factored it correctly. Okay, so the
1:18:0218, guys, here is a bracket. I will put
1:18:04inside it 2 multiplied by 3 power 2,
1:18:07and it has the power 1 minus 4n. And
1:18:10what about 36? We want to factor it.
1:18:12Come on, let's factor it together, 36,
1:18:15the 6 is even. We choose the even
1:18:18factor, which is 2, 36 divided by 2
1:18:21gives me 18, that's correct. Continue
1:18:24then. It’s exactly the same as 18.
1:18:25Divided by 2 is 9, divided by 3 is 3,
1:18:28divided by 3 is 1, so you have 36
1:18:30factored into 2 repeated twice and 3
1:18:33repeated twice. I’m not sure, sir?
1:18:36Get your calculator and calculate 2
1:18:38power 2 multiplied by 3 power 2, you'll
1:18:41find it gives you 36. Thank God, I’m
1:18:43sure so I can continue my problem
1:18:46correctly. Alright, so 36. Here is a
1:18:49bracket, it has 2 minus 3n, okay.
1:18:52Inside it, 36 should be placed here. So
1:18:55what is its alternative? 2 power 2
1:18:57multiplied by 3 power 2. That’s it,
1:19:05my friend. Now we have a factoring
1:19:07journey, so please focus with me. In
1:19:09the name of God, the Most Gracious, the
1:19:10Most Merciful. Initially, the 2 power
1:19:13...it will be multiplied. So the 2 will
1:19:16be multiplied here, and multiplied here
1:19:18. So it becomes 3. 2 by 2n is 4n. 2
1:19:25times negative 1 is negative 2, pay
1:19:26attention guys. Many students do a
1:19:29certain move; don't you ever do it.
1:19:31Watch out, what is it? They go to
1:19:33multiply 2n, they write 4n and then
1:19:35bring down the negative. 1, which is of
1:19:37course wrong. When you multiply this by
1:19:40several terms, it must go into the
1:19:41first term and the second. So this
1:19:43should be the answer. Pay attention to
1:19:45this part. I have to tell you as I go
1:19:47to watch out for certain things,
1:19:48because I know you'll fall into them,
1:19:50so be careful. Well, God willing, He
1:19:52will grant you success and you won't
1:19:53fall into any mistakes. You have here 1
1:19:55minus 4n. It goes down here and here,
1:19:59so it becomes 2 to the power of 1 minus
1:20:024n. Beautiful. Multiply the 3 power 2,
1:20:07wait a moment. You're supposed to have
1:20:091 minus 4n multiplied by 2; let's do it
1:20:11on the side, it's better. I'm afraid
1:20:13you might get confused. 2 multiplied by
1:20:161 minus 4n, 1 minus 4n. The 2 goes in
1:20:21here, 2 times 1 is 2, 2 times negative
1:20:234 is negative 8n. So when this came
1:20:26down, it came down onto the 2. I
1:20:27calculated it for you outside, so it is
1:20:312 minus 8n. Okay, and down below, let's
1:20:34start, in the name of God. This is a
1:20:38power, see? It will descend onto this
1:20:42and onto this. So here is the 2, my
1:20:44friend. And we will do 2 multiplied by
1:20:47this. Let me do it for you on the side
1:20:48so you don't cry. 2 had 2 minus 3n
1:20:51descend onto it. So you will put the 2
1:20:54in here and in here. 2 times 2 is 4, 2
1:20:57times negative 3 is negative 6n. So
1:21:01when this power descended onto the 2, I
1:21:03calculated it outside as 4 minus 6n.
1:21:06Nice talk. Multiply, the 3 is supposed
1:21:09to be here, it will have it descend
1:21:11onto it. It already has a power 2, it
1:21:13will have 2 minus 3n descend onto it.
1:21:14Meaning 2 multiplied by 2 minus 3n. The
1:21:17same as this one, guys, we calculated
1:21:18it once here. 2 multiplied by 2 minus
1:21:213n is 4 minus 6n. God is great.
1:21:26Honestly, it took some effort, but God
1:21:29willing, the problem is correct and has
1:21:31no issues. For me now, I have
1:21:35everything as threes and twos. When you
1:21:38have threes above and below and twos
1:21:40above and below, what do I do, sir? I
1:21:42will show you right away. I write one 2
1:21:45. And one 3. That's it then. I just
1:21:50want to clear some space for myself.
1:21:51Let's take the 2. The 2 first. You are
1:21:56supposed to take what's on top and put
1:21:57a plus before it, and what's below put
1:21:59a minus before it. This is my agreement
1:22:01with you. When you have the 2 existing
1:22:04above and below, any power above, put a
1:22:05plus before it, so 1 minus 4n, I will
1:22:07write it. Now this power is below, put
1:22:13a minus before it, but watch out, when
1:22:15you put the minus, you must open a
1:22:17bracket for me, And put inside it 4
1:22:22minus 6n. Again guys, this part, for
1:22:24God's sake. I did this, I put before it
1:22:26. A plus, see? Meaning I wrote it, okay
1:22:29. Because this is the first one. So the
1:22:30plus before it doesn't make a
1:22:31difference. Here is 4 minus 6n, I wrote
1:22:33it down. I should put a minus before it
1:22:35. But you know, when you put a minus,
1:22:37you open a bracket. Because you will
1:22:39put several terms after it. The 4 and
1:22:41the negative 6n. So these are two terms
1:22:44. If you're going to put a minus before
1:22:45them, there must be a bracket, guys.
1:22:47This is the point I talked to you about
1:22:48in the previous problem. We finished
1:22:50the twos. Let's write the threes. Here
1:22:54is the 3. What are we going to do? 4n
1:22:58plus 2 minus 8n. Step by step. 4n plus
1:23:032, sorry, it is 4n minus 2. This is a
1:23:10plus. This is plus 2 minus 8n. Alright,
1:23:16my friend. Now, you have this power
1:23:19here at the bottom. So a minus must be
1:23:21placed before it. And of course, there
1:23:22must be a bracket. So here is the minus
1:23:25and here is the bracket. 4 minus 6n.
1:23:28Here is 4 minus 6n. Wonderful. Okay,
1:23:33step by step. I want to work on the
1:23:35powers now. I have nothing left but the
1:23:37powers to work on. Here is 1 minus 4n.
1:23:40The negative enters here, becoming
1:23:42negative, and the negative enters the
1:23:45negative 6n, becoming positive 6n. Okay
1:23:47. And here is the three I have. Look,
1:23:50these can be calculated. 4n minus 8n.
1:23:54Well, initially, the negative 2 goes
1:23:55with this positive 2. Initially, just
1:23:57that. And you have 4n minus 8n, which
1:24:00is negative 4n. Good. The negative will
1:24:03enter here on the 4, so it becomes
1:24:05negative 4; it was already there, so
1:24:08when the negative entered, it became
1:24:10negative 4, and I had negative 6n, when
1:24:12the negative entered it, it became
1:24:14positive 6n. Beautiful. Here is the 2.
1:24:19Look, my friend, 1 minus 4 is negative
1:24:223, negative 4n plus 6n is positive 2n.
1:24:28Okay, multiply. Of course, guys, I keep
1:24:30writing the 2 next to the 3, which
1:24:32means there is multiplication between
1:24:33them, in case anyone doesn't know. And
1:24:35here is 3 to the power of negative 4n.
1:24:37Plus 6n. Equals 2n minus 4. I have
1:24:42finished this problem of mine. This is
1:24:44my final answer. This is my simplest
1:24:48form, guys. And thus, I have finished
1:24:50my problem. The next question says. It
1:24:52tells you to find the value of the
1:24:53expression in front of you. And also
1:24:55when the m equals. Step by step, you
1:24:57have to do the simplest form, or
1:24:59someone might tell me, "I will
1:25:00substitute m with 1 here and finish it.
1:25:02" No, you must use the simplest form,
1:25:05don't joke with me, don't be crazy, I
1:25:07mean, what should I tell you? Don't be
1:25:09crazy, surely he didn't bring it for
1:25:11you just to remove m, put 1, and be
1:25:13done with it. You must do the simplest
1:25:15form, okay. So, what are we going to do
1:25:18now? To me, the 5 is small, and 7 to
1:25:22the power of m minus 1 is small too. We
1:25:26want to simplify the 5, let's go. So we
1:25:28said if we have 5 here, we choose 5
1:25:31divided by 5 which gives us 7. For 7,
1:25:33divide it by 3; if I put 3 here, 7
1:25:35divided by 3 won't give an integer to
1:25:38put here, it'll give a fraction, so
1:25:40that won't work. The 30, we said when
1:25:42it doesn't work, take the next one. The
1:25:45odd is not 5, of course, but 7 divided
1:25:48by 7 gives 1. So, 30, guys, is broken
1:25:51down into 5 and 6, meaning 5 multiplied
1:25:55by 6. So, you'll come to the 30 and
1:25:58make it 5 multiplied by 6, all to the
1:26:01power of m. Okay, so 5 power m, over,
1:26:07bring the m in here and here. , so it
1:26:13becomes power .Great. Great, so what do
1:26:22we do now? Be a smart, nice, cool,
1:26:24sweet, and calm kid, and notice that 5
1:26:27power m with 5 power m will cancel out.
1:26:31If you managed to say what I just said,
1:26:33then you are a cool, nice, and sweet
1:26:35kid. You'll put the 7 here. You'll do
1:26:39the top, which is m minus 1, minus the
1:26:40bottom. Of course, I have an m below,
1:26:44so I don't need to open a bracket after
1:26:45the minus, I don't need to. Anyway, the
1:26:47m will go with the negative m, leaving
1:26:49you with 7 power negative 1. Which is,
1:26:53in the end, 1 over 7. He asks me to
1:26:57find out when m is 1; honestly, guys,
1:27:00this is the simplest form. Great, if
1:27:04there was an m left over, I would have
1:27:08gone and replaced the m with 1, but
1:27:11since there is no m here, the answer is
1:27:141 over 7. And by the way, if you went
1:27:17and put 1 in place of m here, it would
1:27:21eventually give you 1 over 7. That's it
1:27:24, boss. Want to verify? Let's go. 5
1:27:31power 1, multiplied by 7 power 1 minus
1:27:341, over, 3 power 1, gives you 1 over 7.
1:27:37We verified it, thank God. Let's see
1:27:41the next question. Pay attention, I
1:27:44have nothing new to say. The next few
1:27:46problems we'll solve are basically
1:27:48similar. So you have 25, break it down,
1:27:51it's clear now how to break down the
1:27:53factors. The 25 becomes 5 power 2, to
1:27:56the power of x plus 1, multiplied by 9,
1:27:59which becomes 3 power 2, to the power
1:28:01of x. Okay, let's go step by step so no
1:28:05one gets confused. The 15 will be
1:28:07simplified and broken down into 3
1:28:11multiplied by 5, all to the power of 2x
1:28:14minus 1. The part on the right, of
1:28:17course, needs to be simplified first.
1:28:19Just handle the left side because it's
1:28:21quite large. So, keep simplifying and
1:28:23organizing things. Then we'll take the
1:28:25one on the right and simplify it.
1:28:26Alright, want to simplify it now?
1:28:27Simplifying it now won't make a
1:28:28difference. Let's simplify it now. I
1:28:30have 375 here, Here's a 5, so let's
1:28:37pick 5. Let's calculate 375 divided by
1:28:405, which gives 75. Here's a 5, let's
1:28:44pick 5, 75 divided by 5 gives 15.
1:28:47Here's a 5, let's pick 5, it gives 3.
1:28:50For 3, we pick 3, the same as it,
1:28:53giving 1. So 375 factors into 5 power 3
1:28:57multiplied by 3. The 5 is repeated 1, 2
1:29:01, 3 times, so 5 power 3 and here 3
1:29:05power 1. All good? All good. So guys,
1:29:08instead of 375......5 power 3
1:29:12multiplied by 3. We want to check, of
1:29:14course: 5 power 3 multiplied by 3 gives
1:29:16375. So, I’m on the right track,
1:29:18thank God. Let's see what we do next.
1:29:24Okay, you have the 2 as a power over
1:29:26all of this. So the 2 will distribute
1:29:29to the x, 2 times x and 2 times 1, so
1:29:31you have 5, 2 times x is 2x, 2 times
1:29:34positive 1 is positive 2. Multiplied by
1:29:393, 2 times x is 2x. I'm multiplying
1:29:43them, okay. This power, guys, will
1:29:46distribute to the 3 and the 5. So 3
1:29:49power 2x minus 1, and 5 power x minus 1
1:29:52, obviously with what between them?
1:29:56Multiplication. What are we going to do
1:29:58? Write the 3 once and the 5 once. Of
1:30:03course, there is multiplication between
1:30:04them. For the 3, you do the ones on top
1:30:07and the ones on the bottom, guys.
1:30:09Anything on top gets a plus before it.
1:30:11Anything on the bottom gets a minus
1:30:12before it. So, in short, I'll do 2x
1:30:15minus, this is the 2x minus all of this
1:30:20. These are two terms, so I must open a
1:30:23bracket: 2x minus 1. So I did top minus
1:30:26bottom. But if there's a minus and
1:30:29multiple terms follow, I must open a
1:30:31bracket. Done. The negative goes inside
1:30:33here and inside here. So you have 3
1:30:36power 2x, negative enters 2x to become
1:30:39negative 2x. Negative enters negative 1
1:30:42to become positive 1. Okay, let's go
1:30:44back to the 5 once more, please,
1:30:46because we hadn't done it. So we do the
1:30:49top, which is 2x plus 2. Minus the
1:30:55bottom, which is 2x minus 1, but we
1:30:57must open a bracket, okay. And here is
1:31:00the 5, I'll bring it down here. The
1:31:02negative will go inside here and inside
1:31:04here. You have 2x plus 2. The negative
1:31:08enters the 2x to become negative 2x.
1:31:10The negative enters negative 1 to
1:31:11become positive 1. Where do we continue
1:31:15? Look, initially, the 2x and negative
1:31:172x will cancel out. So you're left with
1:31:193 power 1, which is 3. Let me write it
1:31:21here, please. And for the 5, the 2x
1:31:25will cancel with negative 2x. And you
1:31:28have 2 plus 1 equals 3, so 5 power 3.
1:31:31Anyway, if you calculate all of this,
1:31:33it will give you 375. Look guys, this
1:31:38question didn't ask you to find x. If
1:31:41it had asked to find x, you might be
1:31:42left with something containing x here.
1:31:44Then you would equate it to the value
1:31:46you simplified. But here he didn't say
1:31:48that, I should have told you from the
1:31:49start of the question. But never mind,
1:31:51he's telling you to "prove that" this
1:31:53equals that. So you had to go and
1:31:55calculate this. We went and calculated
1:31:57it, and it came out to 375. We're done
1:31:59now. You say to him, "Thank God." When
1:32:01I calculated the left-hand side, which
1:32:03is the left part. It turned out to be
1:32:05equal to the right part. So honestly, I
1:32:07didn't have to simplify this or
1:32:09anything. It's not necessary. I have to
1:32:11simplify it to 375, but watch out,
1:32:13because there is another type of
1:32:14question like this that doesn't say.
1:32:17Prove that this equals that. No, it
1:32:18tells you to find x. So you go and
1:32:20calculate this, you keep calculating.
1:32:22You calculate and calculate, and find
1:32:24in the end, for example, that you're
1:32:25left with 3 to the power of x. And on
1:32:28the right, you have 3 squared, so he
1:32:29asks what x is; so x is 2. There is a
1:32:31type of question that might come in
1:32:33this form, like this one, but without "
1:32:34prove that." He tells you to find x.
1:32:37Here, "prove that" means prove that the
1:32:39left part equals the right part. So
1:32:40what do I do? Go calculate the left
1:32:42side, and we calculated it and it came
1:32:44to 375. Is it equal to the right or not
1:32:46? It equals the right, so I have proven
1:32:49to you that the left-hand side is
1:32:50indeed equal to the right-hand side.
1:32:53Now, let's look at the next one. Here,
1:32:56it tells you to find the value of x, so
1:32:59what will we do? Look. You have 3 to
1:33:01the power of x. Multiply by 27, which
1:33:04expands to 3 cubed, and with x on top,
1:33:06it becomes 3x. Right away, then. I've
1:33:08become a fast guy now. 81 expands to 3
1:33:11to the power of four. You expand it.
1:33:13And with x on top, it becomes 4x. Tell
1:33:15me, what is this? Easy, sit down. If
1:33:17you—I am skipping steps. You do it,
1:33:19it's very easy, honestly. Equals, here
1:33:21I have 2 to the power of negative 2x.
1:33:25Let's calculate here, it's all threes.
1:33:26So write a single three. Anything on
1:33:28top, I put a plus before it. Meaning
1:33:30the x, I put a plus before it, okay.
1:33:31It's fine since it's the first one. It
1:33:33won't make a difference, even if I put
1:33:34a plus here. Now I have 3x, put a plus
1:33:36before it. Now I have 4x underneath,
1:33:39put a minus before it. Like that. The
1:33:43important thing is you have 3x plus,
1:33:46plus 3x equals 4x. In short, you do 1 +
1:33:503-4, it will give you 0, okay, equals 2
1:33:54to the power of negative 2x. So I have,
1:33:58guys, 2 to the power of negative 2x
1:34:00equals 3 to the power of 0, which is 1.
1:34:01This is what I've reached so far. 2 to
1:34:05the power of what gives me 1? I know
1:34:07that any number raised to the power of
1:34:090 gives 1, so negative 2x equals 0.
1:34:13This power is negative 2, guys. The
1:34:17negative 2 goes to the other side by
1:34:20what? Say it, say it, don't be afraid,
1:34:22don't be afraid, will I do something to
1:34:23you? Say it goes by dividing. Oh no, I
1:34:27don't want any craziness now, guys.
1:34:31Guys, move the negative 2 over with
1:34:34multiplication. It becomes division on
1:34:37the other side. I know your sneaky
1:34:39tricks, so I tell you negative 2, and
1:34:41you tell me it goes over as positive.
1:34:44No, my dear, this is multiplying the X.
1:34:46So when it moves, it goes over as
1:34:49division. So X equals 0 divided by
1:34:51negative 2. So X. Equals 0. We finished
1:34:55our problem. X equals 0. If you want to
1:34:57verify, go here. Plug 0 in for every X
1:35:00and calculate. You will get a specific
1:35:01number. Same thing, calculate here. You
1:35:05will get the same number. Generally,
1:35:07the number you get here will be 1 and
1:35:09here will be 1, so I thought I'd let
1:35:11you try it with the calculator. That's
1:35:14it, we finished this question. Let's
1:35:15look at the next question. It tells me
1:35:17to simplify. And after you simplify,
1:35:19find the value when X is 1, okay. As
1:35:21for the 32, it needs to be reduced.
1:35:23Let's reduce it together. Here is the
1:35:2532. I'll divide it by 2, it gives 16,
1:35:28divide it by 2, it gives. 8, obviously
1:35:31it's even, so I chose it. I mean, I'm
1:35:33assuming you already know how now. 8
1:35:35divided by 2 gives 4, divided by 2
1:35:38gives 2, divided by 2 gives 1, so 2 ^ 5
1:35:40. Power 5, so 2 ^ 5 and above it is X,
1:35:47so 5X immediately. Because the 5 will
1:35:50be multiplied by the X. Well,
1:35:52multiplied by 3 ^ 2X plus 1, over the 6
1:35:55, we expanded it, which will be 2 times
1:35:583, we expanded it multiple times and it
1:36:01has the power which is 2X plus 1, okay,
1:36:03multiply. Look at this part. You have
1:36:06root 2 raised to the power of negative
1:36:084X. Guys, we learned a rule, let me
1:36:11remind you of it. If I have, for
1:36:13example, square root X squared, the
1:36:15answer is X. Where did it come from?
1:36:18That in square root, guys, there is a 2
1:36:21here, so we are supposed to do the
1:36:23power divided by the root's number.
1:36:25Meaning 2 divided by 2, so it gives X ^
1:36:281, one more. If I have square root X ^
1:36:316, the answer is X. And I do 6 divided
1:36:33by the 2, which is there in the square
1:36:36root, you know. 6 divided by 2, so the
1:36:39answer is X ^ 3, but for example, if I
1:36:43had cubic root X ^ 9, how do I solve
1:36:46that? I do X and I do 9 divided by 3,
1:36:50so the answer is X ^ 3. That's it,
1:36:55that's the way. Okay, sir, how will
1:36:57this help me? I'll tell you. You have
1:37:00here root 2 raised to the power of
1:37:03negative 4X, guys. The power, if it's
1:37:06outside the bracket or even if it was
1:37:08on the 2 inside, it wouldn't matter. It
1:37:10wouldn't matter. Meaning, for example,
1:37:13this, instead of it coming as X ^ 9, no
1:37:15, it's coming with X here and power 9
1:37:18here. Can I, sir, make it X ^ 9 over 3,
1:37:22which becomes X ^ 3? Yes, that works
1:37:25too, normally. There you go, try it
1:37:26with the calculator. Look, if you did
1:37:30cubic root, let's say this X is 2, do 2
1:37:34^ 9, it gave me 8, good. Okay, remove
1:37:39it, look how I wrote it. Look, see how
1:37:44I wrote it, uh. I've written the 9
1:37:47inside. So, what am I going to do now?
1:37:48Stay with me. I will take the 9,
1:37:52instead of keeping it inside, I'll make
1:37:55it a bracket to the power of 9, let's
1:37:57just close the bracket. Press equal, it
1:38:00still came out to be 8, so that's what
1:38:02I wanted to say. To you, it's the same
1:38:04thing. Whether the power is on the
1:38:06outside, I'll still do 9 divided by 3
1:38:08and get the same answer. What is the
1:38:10benefit of this? That I will write the
1:38:12root 2. Okay, no, sorry. I won't write
1:38:15root 2, I'll just write the 2 normally.
1:38:17And its power, which is negative 4x, is
1:38:20here. There should be a 2. Look, look,
1:38:22this is the power here. Which I will
1:38:25divide by the 2 here. So, I wrote the 2
1:38:27that is inside. And I will do negative
1:38:294x divided by 2. That's the power, okay
1:38:33. So I have. Let's write it down one
1:38:35more time. Here is 2 to the power of 5x
1:38:37. Multiply 3 to the power of 2x plus 1.
1:38:391 over underneath. You have x plus 1.
1:38:44It will distribute to the 2, and it
1:38:46will distribute to the 1. Okay.
1:38:51Multiply here, negative 4 divided by 2
1:38:53is negative 2. So it's negative 2x.
1:38:58This is basically the idea you have. Of
1:39:00course, here for You have 3s and 2s, so
1:39:04you will write. But be careful, this is
1:39:05exactly the same as this, so they
1:39:07cancel out. You have 2, 2, and 2, write
1:39:10me one 2. The one on top is 5x, write
1:39:13it down. I should put a plus before it,
1:39:14but it's the first one, so it doesn't
1:39:15matter. Then minus 2x plus 1, minus 2x
1:39:20plus. 1. Okay, there is one more down
1:39:26here. Minus negative 2x, minus negative
1:39:312x. See what I mean? Okay. So it
1:39:37becomes 2 to the power of 5x. The
1:39:41negative will enter the 2x to become
1:39:42negative 2x. The negative will enter
1:39:45the positive 1 to become negative 1,
1:39:47and you have negative negative which
1:39:49becomes positive 2x. That's it. Look,
1:39:52the negative 2x and the positive 2x
1:39:53cancel out. 5x minus 1, this is the
1:39:56final answer, 2 to the power of 5x
1:39:59minus 1. Okay, it says put x equals 1,
1:40:01alright then. I'm pretty much done,
1:40:04here is 2 to the power of. 5 multiplied
1:40:06by 1 minus 1, which will be 2 to the
1:40:09power of 4. Which is 16. The final
1:40:13answer is 16, guys. That concludes my
1:40:15problem. Look guys, we are continuing
1:40:17our review, of course. But I paused for
1:40:18a moment to tell you something. The few
1:40:21questions I'm going to solve with you
1:40:22have some concepts in them. Keep in
1:40:24mind that we have finished the unit.
1:40:27But I included a few questions for you
1:40:29that have some ideas. So, if you've
1:40:31been watching for a long time straight
1:40:33without taking a break. It would be
1:40:35better if you took a short break,
1:40:37especially if you were watching
1:40:39continuously. Take a break, maybe for
1:40:41half an hour. Do anything you like,
1:40:44just enjoy it. And then come back and
1:40:47continue with me. Actually, half an
1:40:48hour might be too long. Take a
1:40:50twenty-minute break. And then come back
1:40:52to me to finish the few questions that
1:40:54have some tricks. I need you to be
1:40:56alert and sharp. I don’t want any
1:40:57silly mistakes, I’m telling you now.
1:40:59Okay, so the few questions I’ll start
1:41:01solving with you now have some clever
1:41:03ideas. They are all essentially the
1:41:05same concept. For example, your first
1:41:09question here is 2 to the power of x
1:41:12equals 5, and you’ll see if I use the
1:41:14Golden Rule. I would say: if the powers
1:41:17are unknown, make the numbers smaller.
1:41:19But these numbers are already small.
1:41:21You’ll tell me, "Sir, let me try on
1:41:22the calculator.""2 to the power of what
1:41:25gives me 5?" You'll do 2 to the power
1:41:28of 1 is 2, 2 to the power of 2 is 4, 2
1:41:29to the power of 3 is 8—wait, there is
1:41:31no specific number. It must be a
1:41:33decimal number. I don’t even want to
1:41:35find it. Rather, in problems of this
1:41:38type, they give it to you as a given.
1:41:41You will use it for what is being asked
1:41:45of you. Again, they give this as a
1:41:49given, okay? You will use it for what
1:41:52is being asked of you. So, the Golden
1:41:55Rule stays. But we apply it as much as
1:41:58we can. I have this 8, which is big,
1:41:59and I need to make it smaller. Wait,
1:42:01this 8 is actually what I want. I want
1:42:048 to the power of x. So I’ll try to
1:42:06figure it out. The 8 will be 2 to the
1:42:09power of 3, all raised to the power of
1:42:11x. Is everything clear so far? You'll
1:42:12say, "Go ahead, sir, multiply the 3 by
1:42:14the x." So it becomes 2 to the power of
1:42:163x. Hold on, stay sharp. I told you
1:42:18these are tricky problems. Stay sharp.
1:42:20Look, my dear. Is there a difference,
1:42:23guys, if I write 2 to the power of 3 to
1:42:25the power of x? Or if I wrote it as 2
1:42:28to the power of x to the power of 3? Is
1:42:31it the same or not? Mr. Mostafa says it
1:42:33is. It doesn’t really make a
1:42:35difference. You have 3 times x here,
1:42:38and you have x times 3 here, and 3
1:42:40times x is 3x, and x times 3 is 3x. So
1:42:43it doesn't matter. Right, so why did
1:42:45you do it that way, sir? I’m trying
1:42:48to structure my problem according to
1:42:50the given provided to me. He gave me 2
1:42:52to the power of x. So I want to have 2
1:42:54to the power of x. So that I can remove
1:42:57it and put what in its place? Five. So
1:43:00I will replace 2 to the power of x with
1:43:025, and the power of 3 is on top of it,
1:43:04so the final answer is 125. That's a
1:43:06good answer, guys. That's a good
1:43:09question idea, okay? From the textbook,
1:43:12let’s see another one. Similar to it,
1:43:16you have 3 to the power of x equals...
1:43:18you don't know anything, so what will I
1:43:21do? I don't know how to do anything.
1:43:23Well, he says he wants you to get this.
1:43:26The Golden Rule says make the numbers
1:43:28smaller, so try...so I will try to pick
1:43:30some rules from here and some from
1:43:31there, but the basics are there, that
1:43:33the Golden Rule is to make numbers
1:43:35smaller, but here I can't do anything.
1:43:38So look at what is required. Because
1:43:40when you get stuck on the given, look
1:43:42at the requirement. So you have 9,
1:43:44which is 3 to the power of 2, raised to
1:43:46the power of x plus 1. Honestly guys, I
1:43:51don't want the two to be the exponent
1:43:53on the three. I want to bring the x, I
1:43:56want the x to be on the what? On the
1:43:58three. Honestly, I’ll do the same
1:44:00trick I did last time. You see this
1:44:03exponent? I’ll swap it with this
1:44:05exponent. Of course, I can’t just
1:44:07swap the x alone and leave the one;
1:44:09it’s not that random. This is an
1:44:12exponent and that’s an exponent; swap
1:44:14them for me. Follow me step-by-step: it
1:44:17becomes 3 to the power of x plus 1, and
1:44:19all of that to the power of 2. I
1:44:21swapped them. But honestly, I also
1:44:24don’t want it to be 3 to the power of
1:44:26x plus 1. I want it to be 3 to the
1:44:28power of x so I can remove it and put '
1:44:30b' in its place. Allow me to take this
1:44:333 to the power of x plus 1 aside to
1:44:35explain something to you. 3 to the
1:44:38power of x plus 1, my dear, if I want
1:44:40to keep it as just 3 to the power of x,
1:44:43I can break it down in a way I'll show
1:44:45you: I'll make it 3 to the power of x
1:44:47multiplied by 3 to the power of 1. What
1:44:52did you just do, sir? Look, if this
1:44:55wasn't here and I gave you these, and I
1:44:57told you 3 to the power of x multiplied
1:44:59by 3 to the power of 1, you would say,
1:45:01"Sir, write the 3 and add the exponents
1:45:03." There it is, write the 3 and add the
1:45:06exponents. Well, I did the opposite. I
1:45:09had this, I had this, and I derived
1:45:11these from it. Okay, you have to
1:45:13challenge yourself. These are questions
1:45:16with concepts, so you might start doing
1:45:18things in reverse, no problem. Keep in
1:45:20mind, I will explain all the concepts
1:45:23that, God willing, the exam won't go
1:45:25beyond. So just stay focused with me.
1:45:28Don't say, "If it comes up, how will I
1:45:30think about it in the exam or how will
1:45:31I solve it?" I am explaining all the
1:45:34concepts, don't worry, the things that
1:45:37involve ideas, I’m explaining them
1:45:39all to you now. So anyway, I've agreed
1:45:41with you now that I’ve broken down
1:45:44the 3 to the power of x plus 1 into 3
1:45:46to the power of x multiplied by 3 to
1:45:48the power of 1. Okay, so I will make it
1:45:513 to the power of x multiplied by 3 to
1:45:53the power of 1, which is 3, that is to
1:45:56the power of 2. The problem is solved
1:45:58for me. Why? 3 to the power of x,
1:46:00I’ll put 'b' in place of all of this.
1:46:03So I put 'b' multiplied by 3, all of
1:46:05that to the power of 2. Of course, 'b'
1:46:07multiplied by 3 is 3b, all of that
1:46:09squared. The power of 2 will apply here
1:46:12and here, so 3 squared becomes 9, and
1:46:14it becomes b squared. This is the final
1:46:17answer, guys: 9b squared. Alright,
1:46:21let's look here. If you look at 2 to
1:46:24the power of x equals 5, if you try to
1:46:26flip things, you won't be able to find
1:46:28x, so it's clear that the one I'll be
1:46:31able to work with is this required
1:46:32expression. It asks me to find it. As
1:46:37for me, I'll break down the 8 into 2 to
1:46:39the power of 3, all to the power of x,
1:46:42minus 3 multiplied by 2 to the power of
1:46:44x plus 2. Okay, we already know a trick
1:46:47; you have 2 to the power of x, so I
1:46:49want to find 2 to the power of x here
1:46:51as much as I can, so we can do a trick
1:46:54and swap these two, so it becomes 2 to
1:46:56the power of x all to the power of 3,
1:46:59minus 3 multiplied by—here, guys, I
1:47:02have 2 to the power of x plus 2. Let's
1:47:04take it aside for a second, you and I,
1:47:07and have a little conversation with it.
1:47:10Like, I want to extract 2 to the power
1:47:12of x from it, just like the last one.
1:47:16You'll make it 2 to the power of x
1:47:18multiplied by 2 to the power of 2. We
1:47:20do it in reverse; if it's 2 to the
1:47:22power of x multiplied by 2 to the power
1:47:25of 2, you write the 2 and add the
1:47:27powers, here is the 2 and I added x
1:47:29plus 2, there it is, x plus 2. I
1:47:31actually had this and reached these
1:47:33because I want to extract from this to
1:47:36have, or to just have 2 to the power of
1:47:38x in it. So, in short, guys, sorry, use
1:47:42a bracket since it's negative 3
1:47:44multiplied by what appears here, okay?
1:47:47Even if you don't use a bracket, it's
1:47:49fine, but let's be organized. The 2 to
1:47:52the power of x plus 2, which is this
1:47:54one, I broke it down into 2 to the
1:47:56power of x multiplied by 2 to the power
1:47:59of 2. Wow, now the problem has become
1:48:01very easy. Since 2 to the power of x is
1:48:055, I will remove this and put 5 to the
1:48:07power of 3 in its place, minus 3
1:48:09multiplied by the 2 to the power of x,
1:48:12I'll remove it and put 5, multiplied by
1:48:142 to the power of 2. You will calculate
1:48:17all this on the calculator very simply,
1:48:19so 5 to the power of 3 minus 3
1:48:21multiplied by 5......multiplied by 2 to
1:48:27the power of 2, I hope. This came out
1:48:31to 65, guys. That's it, next question,
1:48:38same story. You have 3 power x equals
1:48:414. If you were to reduce the 4, it
1:48:44becomes 2 power 2, but that doesn't
1:48:46help because if you have 3 power x
1:48:48equals 2 power 2, the bases aren't the
1:48:51same and the exponents aren't the same,
1:48:54so I still can't get what I need.
1:48:57Honestly, I can't do anything with this
1:49:00one. Let's look over here. The 9 can be
1:49:03broken down into 3 squared. We have 3
1:49:06squared, then power x plus, this is our
1:49:09expansion right here. Look, I have a
1:49:12bracket, put power x in it, multiplied
1:49:14by 3 power...haven't we expanded this
1:49:17several times like this? Then this,
1:49:21I'll make it 1 over 3 power...your
1:49:23problem is now...you just need to swap
1:49:25these. Yes, your problem, after
1:49:28swapping them, becomes entirely 3 power
1:49:31x. Everywhere you have 3 power x,
1:49:34replace it with 4. Allow me to just
1:49:36swap these. Let's erase; it doesn't
1:49:39matter. The 2 will be here and the x
1:49:42will be here, so I've saved a step.
1:49:44What do we do? Everywhere there's 3
1:49:48power x, we replace it with 4, so it
1:49:50becomes 4 squared, plus here 4
1:49:52multiplied by 3...3 power x, guys, I
1:49:55replaced it with 4, plus 1 over 4. Now
1:49:57calculate this very easily. 4 squared
1:50:01plus 4 times 3 plus 1 over 4. Calculate
1:50:04that and it gives 28. Point 25. That's
1:50:10it, that's the solution to my problem.
1:50:15Alright, let's look at the next problem
1:50:19. It says you have 3 power 2x minus 1
1:50:22equals 27. Here you can simplify the
1:50:27numbers, so you have 3 power 2x minus
1:50:301. We've done this many times, 3 power
1:50:323. The base is the same as the base, so
1:50:34the exponent is the same as the
1:50:36exponent. Move this to the other side
1:50:40as positive, so 2x equals 3 plus 1,
1:50:42which is 4. Move the 2 to the other
1:50:44side by dividing, so x equals 4 divided
1:50:47by 2, which is 2. We found that x is 2.
1:50:49Great. Come here, root 7 power x, we
1:50:53found x is 2, so 2 plus y, this equals
1:50:5549. I explained something a little
1:50:58while ago, I told you when you have
1:51:00something like this, consider that
1:51:02there is a 2 in the square root. So you
1:51:07take this power and divide it by 2. I
1:51:10will write 7 raised to the power of 2
1:51:13plus y over 2 equals 49. When we expand
1:51:15it, I did this with you at the very
1:51:18beginning of the review. 49, what is it
1:51:21? 49, guys, is not a number for which
1:51:24you choose 3. If you try 49 divided by
1:51:263, it doesn't work. So you must move on
1:51:30to 7. 49 divided by 7 gives you 7. Not
1:51:33a number for which you choose 3, so
1:51:35that doesn't work, so you take 7 and
1:51:37choose 7 for it. 7 divided by 7 is 1,
1:51:41so 49 becomes 7 squared. Okay, that's 7
1:51:44squared. Look, the base is the same as
1:51:47the base, so the exponent is the same
1:51:49as the exponent, meaning 2 plus y over
1:51:512 equals 2. The exponent is the same as
1:51:56the exponent. The 2 is currently
1:51:58dividing, so it goes over by
1:52:00multiplying, so 2 plus y equals 2 times
1:52:032, which is 4. You will move the 2 to
1:52:05the other side by subtracting. It's
1:52:10positive here, it goes to the other
1:52:12side as negative. If it were attached,
1:52:14it would go by dividing, okay? So you
1:52:16have y equals 4 minus 2. 4 minus 2 is
1:52:19what? is 2. We found Y, and thank God,
1:52:21we solved our problem. Are we done? No,
1:52:23it's asking you for something else. It
1:52:26says find the square root of 5 raised
1:52:28to the power of X, which we found is 2,
1:52:31minus Y, which we found is 2. And wow,
1:52:33calculate it on the calculator, and it
1:52:35will give you 1. Easy, guys, truly easy
1:52:40, really, really. So, you have all
1:52:43small numbers here, and it tells you to
1:52:46solve. Here is 7, and here is X plus 2
1:52:50equals...You can take 7 to the power of
1:52:53X multiplied by 7 to the power of
1:52:56negative X, put them together, write
1:52:59one base, and add the first power plus
1:53:02the second power. The first power plus
1:53:07the second power, so you have 7, and
1:53:10here is X plus 2. This equals, guys, X
1:53:13plus...plus minus is minus, so when I
1:53:16say X minus X, it's 0. Wow, 7 to the
1:53:19power of 0 is how much? It's 1. 7 to
1:53:22what power gives me 1? To the power of
1:53:260. So, X plus 2 is 0, so X equals
1:53:29negative 2, guys, negative 2. Great job
1:53:32, Mr. Mustafa, for spoiling us. So, be
1:53:36careful with this question, there is
1:53:38one similar below it with plus signs.
1:53:43This has a plus, while this one has...
1:53:46they are multiplied, so watch out for
1:53:48the difference between these two
1:53:50questions. When they are multiplied and
1:53:55all have the same base, I write one
1:53:58base and add the powers: A plus A plus
1:54:01A plus A, so 4A equals. 16. Okay, so 2
1:54:08to the power of 4A, and the 16, I'll
1:54:09simplify it to be 2 to the power of 4.
1:54:11I simplified it, we've done it many
1:54:13times. So, if the base equals the base,
1:54:17then the power equals the power, and
1:54:19move the 4 to the other side by
1:54:21division, so A equals 4 divided by 4,
1:54:24which is 1. We solved the problem, a
1:54:26very fun and easy problem. The one
1:54:28after it is completely different;
1:54:31instead of multiply, there is a plus.
1:54:33So, when I have a plus, do I do the
1:54:35same thing? Do I write 2 and add A plus
1:54:38A plus A? No such thing, my dear, no
1:54:40such thing. It's a separate question,
1:54:43guys, please. It's almost unrelated to
1:54:46the unit, as it was there last year, we
1:54:48took something similar. So it's fine,
1:54:51it relates to the unit because it has
1:54:53powers, but I mean they put it there so
1:54:55you get confused and mix up this one
1:54:57with the previous one. Watch out, and
1:55:01keep in mind that this is what comes in
1:55:03exams, not the one before it, so you
1:55:05know. So here, what do I do? Look, my
1:55:08dear, these are three terms, you take 2
1:55:11to the power of A as a common factor,
1:55:13open a bracket, when I take 2 to the
1:55:16power of A, I divide it by 2 to the
1:55:18power of A, it gives me...common factor
1:55:20is what you do, you divide this. So
1:55:24when I divide this by this, it gives me
1:55:27plus, this I divide by this, it gives
1:55:29me plus, this I divide by this, it
1:55:31gives me 1, equals 2. Okay, so 2 to the
1:55:37power of A multiplied by 1 plus 1 plus
1:55:391 is 3, equals 2. Move the 3 to the
1:55:41other side by division. So you have 2
1:55:45to the power of A equals 8. It gives me
1:55:508. It became an easy problem, 2 to the
1:55:52power of A equals 2 to the power of 3.
1:55:54The base equals the base, so the power
1:55:57equals the power, meaning A is 3. There
1:56:02is a big difference between this
1:56:03question and the previous one. I told
1:56:06you the method for this one and the
1:56:08method for the last one, so please
1:56:09focus a bit more with me because there
1:56:11is a question exactly like it. So what
1:56:14will we do? 5 to the power of x plus 5
1:56:16to the power of x, let's take it out.
1:56:19Open a bracket, this divided by this—
1:56:21well, the common factor means you
1:56:23divide this by what you took outside—
1:56:25so 1 plus this divided by this is 1,
1:56:27all of this equals 250. 5 to the power
1:56:31of x multiplied by 1 plus 1 is 2,
1:56:33equals 250. Let's move the 2 to the
1:56:35other side by division. It's the same
1:56:38idea, guys, so 5 to the power of x
1:56:40equals 250 divided by 2, which is 125.
1:56:42We've simplified this many times, it
1:56:45will come out to 5 to the power of 3.
1:56:47The base is equal to the base, so the
1:56:49power is equal to the power, which
1:56:52means x is 3. How easy! Wait, I
1:56:54honestly didn't notice! This is Jerry,
1:56:57of course, after finishing the unit one
1:56:59review, and honestly, this isn't Jerry,
1:57:00it's me and you. We finished the unit
1:57:03one review, let's go to sleep now.
1:57:07Let's go to sleep, I'm dying to sleep.
1:57:10Thank God we finished the unit one
1:57:13review, guys. I am very happy, very
1:57:15happy. Of course, stay tuned for videos
1:57:18like this for every unit. I don't want
1:57:22to say anything else. If you want to
1:57:24inquire about the online lessons system
1:57:25in detail, you'll find all of that on
1:57:27the platform. You can go to the Mr.
1:57:28Mostafa Hossam platform on Google,
1:57:30contact us on WhatsApp, do whatever you
1:57:32want, wait for these videos at your
1:57:33leisure, at your leisure, at your
1:57:34leisure. Do whatever you feel like
1:57:36doing. The important thing is that I'm
1:57:38done now, thank God, and Jerry is happy
1:57:39, and you and I are happy. We finished
1:57:41the unit 1 review. I hope you
1:57:42understood and benefited. This was Mr.
1:57:44Mostafa, the Math Engineer, and peace
1:57:46be upon you and God's mercy and
1:57:47blessings.

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