Indicial Equations (Made Easy)

Added: 2026-05-10

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[00:09:39]So today we are looking at an interesting topic and it's more like what we have looked at previously but we are going a step further. In the last lesson, we looked at indices, right? And we talked about some of the laws of indices. We even applied them to simplify some initial expressions.

[00:10:05]But today we want to go to initial equations that is equations in indices.

[00:10:11]Okay? And we'll be applying those laws that we talked about. Can we remember some of the laws?

[00:10:20]Can we remember some of the laws?

[00:10:25]Okay, let me can we remember some of the laws?

[00:10:36]Let me write some that I can remember here. I think we talked about seven of them.

[00:10:43]The first one is multiplying indices of the same base. Right? So if we say for instance x to the power a multiplied by let me use another variable instead of x let me use y to the power a multiplied by y to the power b since they their bases are the same and we are multiplying we add the power. So this will be y raised to power a + b.

[00:11:14]Remember this one, right? The second one we talked about dividing dividing. So y ^ a / y to the power b we subtract the powers, right? So if you are dividing indices of the same base, you subtract the powers.

[00:11:35]Then we also talked about um something like this y to the power a all raised to power b something like this we said this is equal to y all raised to power a b. So multiplying their powers.

[00:12:00]Then we also talked about like um negative negative um powers something like this y raised to power - let's say - 2 this will be equal to 1 / y raised power 2.

[00:12:20]So to remove the negative from the power, you invert the base of that index. So instead of saying y, we now say 1 / y. Then you can safely remove the negative um power. Good.

[00:12:40]Which other one? We also talked about when the power is zero, right? So for instance, y^0 is equal to 1. So anytime you have the power raised to power 0 it is always equal to 1.

[00:12:56]Maybe 5^0 3^0 10^0 100^0 all of them are equal to 1. Good. And then for the sixth one we talked about fractional um fractional power right something like y let's say y raised to power 1 / 2 this will be the square root of y and then for the seventh one we said also for fractional power but this time around the fraction has a denominator and a numerator so something like let's say y^ 3 / 2. So this will be equal to y the square root of y. Okay, that is for the denominator. Then everything raised to power what? The numerator raised to power 3 like this. So these are the basic laws and these are the things that we'll be applying in today's lesson.

[00:14:03]Okay.

[00:14:06]But there was something else that we would have discussed that is standard form that we didn't discuss last time.

[00:14:16]We might look at that very briefly before we move into equations in indices. Okay, let's look at that very briefly. We wanted to talk about it but there was no time and so I left it.

[00:14:33]Okay. For standard form there's a way we write numbers that we call standard form. It is in this format. Like if I write a number like this a * 10 raised to power let's say n.

[00:14:51]This is written in standard form and a is [clears throat] usually is always between 1 and 10. So from 1 to 10. It should not be up to 10 and it should not be less than one. So a is in between 1 and 10.

[00:15:14]And then this n raised to power n this n must be an integer. What do we what what do we mean when we say integer? Integer is they are whole numbers can be positive or negative. So that's how you write numbers in standard form. where we apply um standard form is when writing the number the weight is like in the normal form will not be convenient.

[00:15:45]For instance, let me give you an example. Let's say I have this number 9 5 6 0 0 0 0 0 maybe additional zero. Now this is normal number way of writing number but this is not so convenient if I write it in standard form it will appear better. So that's where we apply standard form for large numbers like this. So for instance, if I want to write this in standard form, since my a has to be between 1 and 10, it means I will bring there, you know, there's a decimal point after this long number. So I'll bring the decimal point 1 2 3 4 5 6 7 8 9 10.

[00:16:44]I'll bring it here.

[00:16:47]So that this a will now be 9.56.

[00:16:52]Like I said a must be between what? 1 and 10.

[00:16:57]Now this is between 1 and 10. Right?

[00:17:03]this the number of digits before the decimal will just be a single digit. If I stopped here instead of bringing it to this point and I stopped here instead like 1 3 4 5 6 7 8 9. If I stopped here then this will be 95.6 which is not between 1 and 10. So I would not stop here. I would have to move till I get to place the decimal point so that the number before the decimal point is just a single digit. Do you get that? So it must always be here. The number before the decimal point must be a single digit.

[00:17:55]So 9.56* 10 raised to power something. So what will I put as n?

[00:18:03]It depends on how many times I moved the decimal point. So from here, let's go.

[00:18:09]Let's count. 1 2 3 4 5 6 7 8 9 10. So this will be raised to power 10. Okay. Now the as I said the number the integer can be positive or negative.

[00:18:30]It is positive when you are moving towards like you are moving the decimal point to the left direction.

[00:18:39]But if you are moving to the right direction that power will be negative.

[00:18:47]For instance, let's say we want to make write this as in standard form 9.

[00:18:57]Which example will I give? Let me see.

[00:19:01]Okay, I think I see I have an example here. 0 0785.

[00:19:10]I want to write this in standard form.

[00:19:14]Okay. So just like I moved the decimal point for the that whole number this large number I moved from here in the left direction.

[00:19:26]This time around the decimal point is here. I'll be moving to the right direction. 1 2 3 4 5 6. So I will stop here because now it will be 7.85 85 which make the number of digits before the decimal to be just a single digit and it satisfies this um rule here for standard form that a must be between what 1 and 10. So it satisfies that rule and so we have it as 7.85* 10. So what will I put as the power?

[00:20:09]I'll check how many times did I move the decimal point.

[00:20:13]1 2 3 4 5 6. So I moved it six times. So raised to power 6. But since I moved it in the right direction, it will take a negative sign. So 7.85 * 10^ - 6. So you can see that between these two, this one and this one, this looks more decent like it looks better than this one writing with all these plenty zeros here. So this one looks is a better presentation and a more standard way of writing numbers like this especially for very small numbers like that has so many uh decimal points or those that are very large that has maybe many zeros and all.

[00:21:09]So this is the standard way we write numbers in mathematics.

[00:21:13]In sciences generally if you even if you go to physics this is the way they will write it. If you go to chemistry they write it this way even biology. All right. So this is a the standard scientific way of writing big numbers or very small numbers. Do you get that?

[00:21:34]Good.

[00:21:36]All right. Maybe you are thinking this is not an important topic. Let me bring some questions from JAM and WK on standard form. So you attempt this for me. Question number one, express 55120 in standard form. This is a WK question.

[00:21:57]So let me go to the comments. Who is expressing 55120 in standard form?

[00:22:13]like I said. Okay. So is is someone doing that [snorts] 55120 in standard form right um Sabira says 5.5120 * 10^ 4 you've cleaned You've canled it.

[00:22:54]Who is answering five 120 in standard form? Okay. Isabel says 5.5 * 10 raised to power 4.

[00:23:06]Really? Why did you take out the 1.1 and 2?

[00:23:14]You're correct. But one and two where did it go to?

[00:23:19]You're not exactly correct. Okay. Um says lame day says 5.512 * 10^ 4. That is correct. Awesome.

[00:23:30]Awesome. Awesome.

[00:23:33]That's awesome.

[00:23:41]Who else is answering?

[00:23:44]Um, Sabira says 5120 * 10^ 4. That is correct. But you don't need to put the zero after the decimal at like zero at the end of a tail end of a decimal point means nothing. So you can remove that zero. Do you understand?

[00:24:07]So the better way to write it is 512 * 10^ 4.

[00:24:14]All right. Um Ad Morium says 5.512 * 10^ 4. Good. I think we are good. GD says 5120 * 10^ 4. That's correct. But like I said, the zero after the tail end of the decimal is not necessary. Okay. So this is the way we did it in case you don't know how to do it yet.

[00:24:43]So we there's always a decimal after after any number but we don't write it for whole numbers. You don't need to write it. So but if you are bringing in you're introducing this um standard form you introduce a decimal point and then move the decimal point 1 2 3 4. So it comes here. This will be 5.512.

[00:25:11]Then time 10 raised to power what? The number of times you moved it which is four times. And since we moved it to the left direction it will take a positive sign. So 5.512 * 10^ 4. Good. Now do this. We just have two more questions on standard form and we'll move on to today's topic.

[00:25:35]Simplify this and express your answer in standard form.

[00:25:40]Simplify this and express your answer in standard form.

[00:26:02]Who is doing it? Who is doing it?

[00:26:18]So what you do is multiply that and then who is doing this?

[00:26:37]Okay, let me do that for us while I because this is not our main lesson for today so that we don't take all the time. So what I will do is to write them individually as standard for in standard form. So I'm moving this 1 2 3 4. So this will now be 2.15 * 10^ what - 4 in standard form time I'm moving this 1 2 3 4 5 okay so this will be 2.8 8 * 10^ - 5. Good.

[00:27:19]So after that I can multiply this one by this one. Okay. 2.15 * 2.8. What would that give us? That will give us 6.02.

[00:27:35]Then* 10 raised to power -4 * 10^ -5. Remember when you are multiplying indices of the same base what do you do to the powers you add. So -4 + -5 that will be - 9. So this is our answer 6.02 * 10^ - 9.

[00:28:05]Okay. Okay. Yeah. Someone got it. Got it. 6.02 * 10^ - 9. You said 9. No, it's - 9. Okay. Um, Sabira says, "Yeah, 6.02 * 10^ - 9." All right. So, G says, "Oh, sorry. It's - 9." Okay, that's correct.

[00:28:25]Gidd or Isabel says 6.02 * 10^ - 9.

[00:28:28]That's correct. That's correct. All right. So, we are doing well. And finally, for standard form, multiply this and leave your answer in standard form.

[00:28:40]Let's see who gets it first.

[00:28:44]Multiply that and leave your answer in standard form.

[00:28:54]I'm waiting who gets it first.

[00:30:38]Okay.

[00:30:43]1 7 * 10^ 3 1.7 * 10^ 3 who's that is um morium that is correct that is correct 1.7 * 10^ 3 is correct Okay, I can see um Isabel says 1.7 * 10^ 3. That's correct.

[00:31:24]Um Sabira says 1.701 * 10^ 3. That's even more accurate.

[00:31:32]Well done, Sabira.

[00:31:35]Muhammad Abdullah says 1.703 * 10^ 3 01 * 10^ 3 that is absolutely correct okay I think we are good with standard form so let's now move on to um indices equation in indices are we ready we ready equations in indices.

[00:32:10]So we've um listed out some laws of indices and we are going to be using them in solving equations in indices. Let's start with this equation number one.

[00:32:28]Solve for a positive number of x such that 2 to the power x raised to power 3.

[00:32:37]Oh no, let me start with a simpler question. Sorry, let me start with question.

[00:32:45]Yeah, this one is simpler. If 9 raised to power 2 - x is = 3 want to find the value of x.

[00:32:58]We want to find the value of x.

[00:33:01]So what would you do? Anytime you are solving equations in indices, this is what you have to bear in mind. You try to see if you can make terms on either side of the equation to be indices of the same base.

[00:33:16]Now why do you need to do that? If you can do that then if the bases are the same the powers have to be the same right? So the only thing that is equal to for instance let me say I have um 5 raised to power two on the right hand side. This is equal to I succeed to make the base to be five.

[00:33:42]Then the only thing that is equal to 5^ 2 must be 5^ 2. There's nothing else.

[00:33:49]Anything you put here that is not two is wrong. So if you can make the b to be of to be the same then the powers if this is x then x must be equal to 2. If this is um if this is x - 3 then x - 3 is equal to 2. Do you understand what I'm saying? So once you can make the base to be the same, then the powers have to be the same. So let's do this.

[00:34:22]9 and three. Can we make them to be of the same base?

[00:34:26]9 and 3. Yeah. 9 is 3 to power two. Is that not so?

[00:34:32]9 is 3^ 2 and 3 is 3 to power 1. So they are now of the same base. But here we have 2 - x. So in brackets we have 2 - x. Okay.

[00:34:50]So we have expressed 9 ^ 2 - x as 3^ 2 then bracket 2 - x. Do you understand?

[00:35:00]Now since the bases are the same three and three then it means the powers will be the same. That means 2 bracket 2 - x is equal to what? 1. Since the bases are the same, the powers have to be the same.

[00:35:22]Awesome. Awesome. Okay. So, I'll open the bracket. Two will multiply two. It will also multiply - x. So, we have 2 * 2 is 4. 2 * - x is - 2x. This is equal to 1. Right?

[00:35:43]Now to get x, I would collect like terms. We have - 2x. I'll move four to the right hand side of the equation. So as it crosses the equal sign, it becomes -4. So we have - 2x is equal to 1 - 4 is - 3. to get x I'll divide both sides by min -2 right so that x is what x is 3 / 2 or you can change it to um mixed number which is one whole number 1 / 2 are we following let me see okay some of us got the answer is one whole number one over two that's awesome that's awesome Um GD which one are you answering? You say X is equal to 4. Really?

[00:36:44]Not really. Not really. X is not equal to 4.

[00:36:52]Okay. This says X= one whole number 1 / two. It's correct. Isabel says one whole number one over two. Okay. Okay.

[00:37:03]Good. So I hope we follow to this points. If you follow to this point says followed. If you follow to this point says followed.

[00:37:20]If you follow to this point says followed right um I mean uh is that okay James Lemur James says followed okay if you follow to this point says followed all right we followed okay let's move on I'll bring another question it's also simple the question is if 9 raised to power 2x is equal to 1 / 3 in bracket 27 raised to power x. We want to find the value of x. This is sce 2009.

[00:38:16]Why I'm putting this is so that you would come to the point where you are not afraid.

[00:38:23]Some persons they do well in class but when they go to exam they feel because they they like this is wo and that's um fear that they are entering they entering into um you know final exams and all they go there and fail. Some people repeat maths. People that have gone through SS1, they didn't fail maths. They went to SS2, they didn't fail. They went to SS3, they didn't fail. They go to WK and they are reciting because of maths.

[00:39:01]Five times they are still writing WK because of maths.

[00:39:07]I'm showing you that all the questions have the first one I asked you both the ones of standard form they are WK and JAM.

[00:39:14]So if you can solve those questions, why can't you solve them in the exam hall?

[00:39:20]That's what I just want to put in your mind so that you don't go into any of these exams afraid.

[00:39:28]In fact, many of the questions I will ask you will be jam which means for someone who is taking WK those questions are a bit higher you know than because JAM is a bit more like tougher than WK so those questions are a bit higher than even what you would you expected of if you are not taking maths in jam so if you can solve those question I don't see any reason why you would struggle in your final exams.

[00:40:05]Okay?

[00:40:06]So this is to build confidence in you.

[00:40:09]That's why I'm writing all this to show you that it's from final exams. Okay.

[00:40:15]Cuz I'm talking I believe you are solving.

[00:40:21]Okay. Yeah. We are following. Has anybody solved this one or All right, let me help you. But then from next one you'll be helping me.

[00:40:33]Okay. 9 to the^ 2x is equal to 1 / 3 27 raised to power x. Remember as I said that if you are solving equations in indices you try to see if you can make terms on either side of the equation to be of the same base. That is one rule you must remember. Anytime you come to solve you see this is an index equation always remember can I make them indices of the same base.

[00:41:04]If you can you are most likely going to get the answer. Okay. So we have 9, we have three, we have 27. We can all they can all be expressed as indices in base 3. 9 is the same thing as 3 to the power what? 2. Then we have this 2x. So times 2x 1 / 3.

[00:41:27]You know when you are 1 / 3 3 is same thing as 3^ 1 right? 3^ 1 like this. But if you want to write it as like three instead of 1 / 3 then you have to introduce a negative.

[00:41:46]Remember one of the laws that we looked at is negative uh power. If you want to remove the negative then you have to invert the base. This will now become 1 / 3. Right?

[00:41:57]But if it is a fraction like this and you want to make it like instead of 1 / 3 you want to make it three then you have to introduce negative.

[00:42:10]Is it clear? Good. Then 27 is 3 to power 3. Then we have this x.

[00:42:19]So now you have expressed it as all of them all the terms as indices of the same base. But on the left hand side we just have three only one three 2 * 2x is 4x.

[00:42:33]But on the right hand side we have three and then three. So we have to combine them. How do you combine them? you are multiplying. If you're multiplying indices of the same base, what do you do to the powers? You add. So we add the powers. 3^ -1 plus the power of this one is 3x.

[00:42:56]Good. So now on either side of the equation we just have three and then three. So the bases [clears throat] are the same. Now we can equate the powers.

[00:43:07]We can say 4x is equal to what? -1 + 3x.

[00:43:17]Now a very simple equation to solve. So what I'll do is to collect like terms.

[00:43:22]I'll take minus I'll take 3x to the left hand side of the equation.

[00:43:28]So it will be 4x as it's crossing the equal sign becomes - 3x. Okay. This is equal to minus1. 4x - 3x is what? Is x.

[00:43:40]So x is equal to what? -1.

[00:43:45]Let me see if we're following.

[00:43:48]Okay. Someone already got the answer.

[00:43:51]Someone says the answer is x = to minus1.

[00:43:56]Correct. G says x is equal to minus1.

[00:43:59]Awesome.

[00:44:02]Lima James says x= minus1. Chidima says or Isabel says x is equal to minus1.

[00:44:12]Awesome. Awesome. Good, good, good. I love this class.

[00:44:17]I love this class.

[00:44:21]Okay. Sabira says x is equal to minus one.

[00:44:26]Okay.

[00:44:28]Awesome. Awesome.

[00:44:31]Okay.

[00:44:32]This class is good. I love this class.

[00:44:37]Alani says x is equal to minus1.

[00:44:42]Okay, dockers. Docker says x is equal to minus1. Good. What an awesome class. All right, question number three. You guys will do this one for me.

[00:44:57]It's is looking a bit more complex but it's not really that complex. So try it for me. Okay, this is it. 5 given that 5 raised to power n + 3 over 25 raised to power 2 n - 3 is equal to 5 raised to power 0.

[00:45:26]We want to find the value of n.

[00:45:30]Find the value of n. This is SSCE 201.

[00:45:38]Sce can do this for us.

[00:45:58]I'm in the comments. I'm waiting for who gets it first.

[00:46:25]Please be looking at the comments. Don't be distracted with the comments. Just when you are solving, when I'm solving or teaching, you look at the screen, okay?

[00:46:39]The video. Then when um when you it's time for you to type in your answer then you can go to the comments so that you'll not be distracted. Okay.

[00:46:57]Who is getting the answer for us? N= what?

[00:47:04]Someone says n = -3. Is it is it really minus3?

[00:47:12]All right, we'll check that.

[00:47:15]Miam says x =us3 says x n = 3.

[00:47:23]Awesome. Awesome. All right, but we still check it. Isabel says n=us3.

[00:47:31]Okay, let's see if it is correct.

[00:47:39]says n =us3 that is awesome okay says n= minus3 g says n is = 3 okay we are getting two answers three or minus3 we are going to confirm which one is correct all right let me do that for us let's do it together let's go there. Okay. So you know that anything raised to power 0 is what is one. So we can easily say that this one that is 5^ 0 is what? 1.

[00:48:20]So on the left hand side we have 5 ^ n + 3. Then this denominator and on the left hand side we can take it to the right hand side. And anything that is multiplying one is still that thing. Right? So this let me just write it here. This is 25^ 2 n - 3. It will come here. So as it's multiplying one, it will still be that thing. Right? So we have 5^ n + 3 = 25^ 2 n - 3.

[00:48:58]Right?

[00:49:00]Now can we express five and 25 as indices of the same base? Yes, they they can both be expressed as indices in base 5. This is already in base 5. So 5^ n + 3 then 25 is 5^ 2 5^ 2. Then we have this 2 n - 3. Right? So now we have them the same base five and five. So what do we do to the powers? We equate the powers. So since the bases are equal, powers have to be equal. So we say n + 3 = 2 bracket 2 n - 3. So now I can open the brackets. 2 we multiply 2 n. It will also multiply minus 3. So we have n + 3 = 4 n - 6. Okay. So what do we do now?

[00:50:03]We collect like terms from here. I'm sure all of us will be able to solve it. So we collect like terms. N I'll take minus I mean 4 n to the left hand side and it crosses the equal sign becomes -4.

[00:50:20]This is equal to -6.

[00:50:22]As + 3 crosses the equal sign becomes what? -3.

[00:50:28]So we have n - 4n is what?

[00:50:32]Is - 3 n - 3 n is equal to - 6 - - 6 - 3 is - 9.

[00:50:48]I don't know maybe this where you guys started making mistake. Okay. So to get n I'll divide both side by minus3 so that n is equal to what? Minus will cancel minus 9 / 3 is 3. So n is equal to three.

[00:51:06]So some of us got it. But those that got minus 3 you also did very well. But you know if this is an objective there will be an option that says minus3.

[00:51:18]Okay there will be an option that will say minus three then there will be one that will say three you know and so because of that it's likely that all your efforts will now be like a waste. Do you understand what I'm saying? So you have to be mindful of sign. Imagine that you solved and you solved and you almost got the answer and instead of putting three you put minus three and then there's an option that is minus3 then it means that all this thing that you did you didn't get reward for any of it because of a simple sign. So be mindful of your signs. Okay, I have uh a few more questions now. Let me go to JAM.

[00:52:10]Uh as I said, you will see that um these are things that you can easily do. It doesn't matter if it's jam or not. So the question is if 27 raised ^ x + 2 / 9^ x + 1 = 3^ 2x we want to find the value of x. This is UTME 20.

[00:52:49]UTME 2012.

[00:52:56]So find me the value of X. I'm sure we can do this. Just take your time and find me the value of X.

[00:53:20]Take your time and find me the value of X.

[00:53:24]Who gets it first?

[00:53:26]And remember, as I said, be mindful of what? Your signs. Be mindful of signs.

[00:54:00]As I said, don't be distracted by the comments.

[00:54:05]Don't be in the comments looking at reading comments. You will not pay attention to this lesson.

[00:54:13]This is the way to go about live class.

[00:54:16]You look at the video when I'm teaching and then when I'm not when you want to put in your answer, you go to the comments, put in your answer and then focus back on the video. That's how to get the best from this. If not, you will be distracted.

[00:54:40]Who's getting the answer for us?

[00:54:53]Oh, GD says four.

[00:54:56]Awesome. Gay.

[00:54:59]Awesome. Wow, that is nice. That is nice. That is nice.

[00:55:06]G says the answer is four. Okay, says the answer is five.

[00:55:12]Okay, let's see.

[00:55:15]Allow me. This says the answer is three.

[00:55:18]Okay, let's see.

[00:55:28]Allow says the answer is four.

[00:55:32]That's good. That's good. That's awesome. Correct. Correct. Correct. This class is amazing.

[00:55:49]Okay, let me let me do it first. Let's do it together. Okay, we have this. Like I said, we try to see if we can make them indices of what? The same base. So, let's try 27. We can write 27 as an index in base 3. 27 is 3 * 3 * 3 which is 3^ 3 then we have this * x + 2 right then divide by 9 we can write as an index in base 3 9 is 3 * 3 which is 3^ 2 then we have this in bracket x + 1 then this is already in base 3 so 3^ 2x Good.

[00:56:36]Now when you are dividing indices of the same base, what do you do to the powers?

[00:56:42]You subtract. So here we have base 3 base 3 and we are dividing. So we subtract the powers. This will be 3 to power 3 x + 2 - 2 bracket x + 1. So this one minus this one.

[00:57:05]This is equal to what? 3^ 2x.

[00:57:09]Now we have them on either side of the equation to be indices of the same base.

[00:57:16]See three and three.

[00:57:18]You don't equate at this point because we have two three here like this one and this one and we just have only one here.

[00:57:27]The time that you equate the powers will be you have one on the left hand and you have another one on the right hand. Do you understand? So now we can equate the powers. It means that this power is equal to this power.

[00:57:42]Awesome. Awesome. Okay. So we have 3 x + 2 - 2 x + 1 = 2x.

[00:57:54]So from here it's very easy to solve. We just open the brackets. 3 will multiply x. It will also multiply what? 2. 3 * x is 3 x 3 * 2 is 6. Then -2 will multiply x. It will also multiply 1 - 2 * x is - 2x - 2 * 1 is -2. This is equal to 2x.

[00:58:17]So now we collect like terms. 3x - 2x is what? X. So we are done with this and this.

[00:58:28]Let me move this this 2x to the left hand side of the equation. So as it crosses the equal sign becomes what? - 2x.

[00:58:38]So we are done with this. Now this is equal to what? Let me move the numbers to the right hand side of the equation.

[00:58:46]So starting with this, let me start with this minus2. As minus2 crosses the equal sign, what does it become?

[00:58:54]It becomes +2. As + 6 crosses the equal sign, what does it become? It becomes -6.

[00:59:03]Good.

[00:59:04]You're making progress. So, x - 2x is what?

[00:59:10]It's - x.

[00:59:14]2 - 6 is what? It's - 4. So, minus will cancel minus x is equal to what? Four.

[00:59:25]Awesome. Awesome. Awesome.

[00:59:28]Awesome.

[00:59:30]Yeah, GD, you're correct again. Keep it up. Okay, that's good.

[00:59:37]That is good. I think that is it. Um, I have more questions, but because of time, I would leave it. But maybe I'll just ask it as um an assignment.

[00:59:51]Possibly next week, I will get the answer from us.

[00:59:56]So are you ready? Let me type them. Let me write it as an assignment. So just write it somewhere.

[01:00:03]Um so you are going to solve for the positive value of x. Solve for the positive positive value of x. And this is the equation 2 ^ x raised power 3 - x^ 2 - 2x = 1.

[01:00:43]Solve for the positive value of x.

[01:00:48]Okay.

[01:00:52]Have you copied it?

[01:00:59]Have you copied it?

[01:01:01]Let me give you one. This one is like um a polomial equation.

[01:01:09]So you know what to do. Okay.

[01:01:13]Another one is this. If five to the power x + 2 y = 5 and 4 to the power x + 3 y = 16.

[01:01:39]you I want you to find 3 to the^ x + y. So do these two and in the next class I will get the answers from us. Okay, I'll ask us what our answers are and then I would if I see that we still need more explanation on it, I will do that. So this one will be more you would apply simultaneous equation because we have two variables.

[01:02:08]Have you done that? Have you copied it?

[01:02:11]Or should I put them together? Let me see. Let me put them together.

[01:02:19]Okay.

[01:02:23]So, this is question one and question two for assignments.

[01:02:28]Question one and then question two. All right. I think that's it for today's class. How was it? How was today's class? Let me know.

[01:02:38]in the comments.

[01:02:41]How was today's class?

[01:02:54]How was today's class?

[01:03:01]All right, someone says today's class was excellent.

[01:03:17]How was today's class? Let me know. Let me know. Awesome. Muhammad Abdul Abdullah says the class was awesome. All right.

[01:03:35]interesting and brain testing. Okay, says the class was interesting and brain testing. Shi says the class was amazing.

[01:03:44]Okay, Chima says the class was superb says thank you sir. Okay, Miam says it was very interesting. All right, I also enjoyed the class and I think that is it for today.

[01:04:02]So remind me in our next class that I said I was going to explain this or ask for this particular assignment so that I will not um I will not forget. Okay, six. That's six and uh five.

[01:04:20]I'll explain I'll explain it to us again in case we need more explanation on that.

[01:04:27]All right, that's it for today.

[01:04:30]You guys were awesome. I love you all and bye-bye.

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