 The things that we also learned last week is powers. So we're also going to also include the powers in our communication and discussion. When it comes to powers, only when it is multiplication and multiplication and division, then we rely on the basis. If the bases are the same, you're either going to add or subtract the powers. If we're dealing with addition and subtraction, then we need to talk about like terms. And the like terms are the things that looks exactly the same or similar to one another. And therefore we can add the like terms together. We can subtract the like terms together as well. So let's look at this question number one. It says simplify the following expression as far as possible. So therefore it means simplify this until there is no way you are able to simplify it any further. And one thing for sure that I forgot to do is to record the session. And I see that someone already started the recording on our behalf. I don't know who started the recording, but I hope it's Adele. If it's any of the students who started the recording please let me know because just let me know who started the recording because we will need to have access to that recording. Okay, otherwise let's continue. So now always when you solve expression also, remember the BOTMAS rule, right? BOTMAS says brackets first before addition, subtraction, exponent or powers and addition and subtraction and multiplication and division. That's what we need to do first. And then we do expressions and then we do division and multiplication. And always remember that division and multiplication have the same priority if they appear on the same equation or formula we work from left to right. Also addition and subtraction have the same priority we work from left to right. So yeah, we have six A squared minus nine AB plus two B squared minus three times B A minus B squared. We first need to get rid of the bracket. So to get rid of the bracket, we are left with six A squared minus nine AB plus two B squared. And we're going to multiply with the negative sign that is in front or the minus that is in front of three. So we say minus three times three B A. We're going to also rearrange B A and start with A because letters of alphabets say A, B, C, D. So we just want to rearrange that as well as we work through. So this will be minus three times B A will be minus three AB. I'm just rearranging B A by writing A first. Next, we need to multiply by minus three into minus B squared. So minus three times minus B squared, negative times negative becomes positive and the answer will be three B squared. Now we need to look for like times. In terms of this, six A A squared, it's on its own six A squared. I'm just going to rearrange this so that I put the like times together. And then we have minus nine AB and we have minus three AB so that I bring the like times together. You don't have to do this step. Save time in the exam. I'm just showing you some of the ways that you can do to identify things quickly. B squared plus three B squared. Or you can also just highlight them and answer the question. So this will be six A squared minus nine AB, minus three AB because they are the same. We can add or subtract. So minus nine, minus three, it's minus 12. AB and plus two B squared, plus three B squared. They are like times. So plus two, plus three is plus five B squared. And the answer is option number two. Answer will be option number two. And that's how you're going to answer questions when they give you expressions and ask you to simplify them to their simplest form. Let's look at another expression unless if you have any question. Remember also the sessions are interactive. It's not going to be me only talking. At some point you will have to unmute and talk to me, right? And if you have any question while I'm explaining things and you're getting lost, just unmute and let me know and ask your question so that then we don't leave anyone behind. Okay, so let's look at exercise two. Simplify the following expression as far as possible. So here as well, we are given an expression of four times the root of 64 squared, 64 X to the power of 16. Now, the other thing you need to also remember when it comes to the roots or the squares and all that you can also simplify by changing the root to a power. So what do we mean by that? We know that a root which is also the same as the square root. So a root without a number in front is the same as the square root. If the square root is written like this, we can also write the square root as a power. So we know that the caravan is the same as one over as the caravan and the number in front of the caravan is what goes in the fraction. And this we can write it as the power. So because I ran out of space there, this square root is the same as a number to the power of, sorry, I wrote it wrong actually as well. Right there, it should be two. The square root can be written as a value to the power of a half. So if let's say here it was an A, therefore it will be A to the power of a half. So if it was the quadro root of B, we can write this as B to the power of one over four. What happens when it is the quadro root? I'm gonna use quadro root of B to the power of a half. You also still need to say this will be one over, sorry, because our B is to the power of A half. And sorry about my flicking chart to the power of one over four. That's all what you need to always remember. A cube root, a square root, a quadro root, whatever root that you will have, you can always say two as the power of that root of a fraction. So coming back to our put equation here. Now we can also break this equation to make it easy for us to understand and unpack it into different parts. So we're going to say four multiplied by the root of 64. I'm going to break everything that is underneath the root into different parts, because a square root of a number, we can find it using a calculator. And the next one will be the root of X to the power of 60. Now, because I have three different equations, I can then just say four multiplied by, what is the square root of 64? What will be that number that when we multiply that number by itself two times, it will give us 64 and that number is eight. Yes. And multiply by, and this we can change it to X to the power of 16 to the power of one over two. Now, because we have X to the power of 16 times the power of a half, we can take 16 multiplied by one over two, right? Two goes one time into two and it goes eight times into 16, eight times one is eight over one, which is the same as eight. That's one way of answering the question as well. So we can simplify it that way. So that will be four times eight. It's how much? 32. It's 32 and X to the power. And we said everything at the power we solved it, it's X to the power of eight. And the answer is option number two. Let's see if you are able to answer this question. So here is your question, question three. I'll give you three minutes. Are we winning? Yes, ma'am. I put my answer in the chat. I see that. Others, are we winning? Are we getting there? Or are we lost? They say asked and it shall be given. So if you ask back and you don't know how to answer it, please ask, ask for help. Yeah, so let's see how we answer it. Now it's your chance to speak so that I can be quiet. See that there are. Please don't forget to complete the register. I see Adele says you must also put your student number on the chat. I think for those who are unable to see the chat, then it might be a little bit difficult. Okay. So who wants to do this? Anyone? Okay ma'am, I will go. Next. Okay, so what I did was I took what is in the brackets first and obviously because it's to the power of two, I then said the P to the power of two then becomes P to the power of eight. And then I took the one over three and I just put it to the board. Like I'm saying, so I said six times one over three, P to the power of eight gives me eight, two P eight. But I also looked at it again and I also saw if I take what's in the bracket like the one over three and I also times that by two. Like if I take the whole number that's in the bracket and I times that by two, it still gives me, like for example, it will still give me the same answer because it's the same principle that's being applied. Okay. So let's do it step by step so that we don't get confused. So we first need to get the bracket. Remember, Otmas is our friend. Brackets first. So brackets, there's nothing we can do with the brackets because there's nothing inside the bracket that we need to work with. The next one is exponents or powers which are represented by O which is the next thing. So the next thing is for us to simplify the exponents. So to do that, we say six, multiply by and we're going to distribute two into the whole bracket. Three to the power of two, multiply by P to the power of four times two which is to the power of two. And what do we have now? You also need to go back to the powers and understand the principles of the powers. So we say in terms of the powers, if I have X over Y to the power of N is the same as X to the power of N divided by Y to the power of N, you can distribute that power into all the values that you have in your fraction. So we can do the same. So one over three to the power of two, we can write it as six times one squared divided by three squared multiplied by and I can already solve this side. It's P to the power of eight. Now we are left with six times what this one squared is one over three squared. What is three squared? Three squared is three times three which is equals to nine times P to the power of eight. Now six times one or we can simplify. Remember, there's a fraction multiplying. So we can also do the following. We can treat six over one times one over nine by saying what is the number that can divide into six and three or simplify into six and three? That is a three. Three goes into six, two times and it goes into nine, three times and the answer here will be two times one will give us two, one times three will give us three and therefore the answer for this question will be two over three P to the power of eight which is option number one. And that's how you're going to simplify expressions to their simplest form. Are there any questions? If there are no questions, then now let's move into... Sorry Elizabeth. Yes. For myself, I don't know if it's going to cause confusion for me when I'm busy with the exams because what I normally do is I would simplify it using this second step and then I'll just calculate the numbers with the calculator. Yeah, it's fine. If you're using a calculator, you can use your numbers. The numbers you can use your calculator, you don't have to do it manually. So you can take your calculator and calculate six times one over three to the power of two, it will give you the answer. I'm doing it manually because I don't have my... I don't have a calculator in front of me and we're not working with big numbers that will require me to use a calculator. But yeah, in the exam, go ahead, use a calculator. So now you're not dealing with fractions, we're going to equations. To solve equation, the purpose of solving equation is to find an unknown number, is to find an unknown number, possibly. So for example, if you're given an equation with only one variable, which means the number we don't know because it's a placeholder, it's a number that we all don't know what that number is unless we calculate that number. And the simplest way of it is to solve that equation to find the missing number. So when you solve equations, I've already in the beginning, when we started, I've told you, whatever you do on the left, you must do on the right, whatever you do on the right, you must do on the left. Or when you move things across the equal sign, the sign will change. If it was minus, it will be plus, if it was plus, it becomes minus. If it's dividing, you multiply. If it's multiplying, you divide to get rid of that number. Those are the principles that you always need to remember when solving equations, not also forgetting about mass. But I can't say for exponents and expressions, division, multiplication, they've got the same priority. We're working from left to right, but they have priority over addition and subtraction. And addition and subtraction have the same priority. We also work from left to right, irregardless of the equation that we have. So when we have equations like this, we need to distribute the value outside of the brackets, because if x plus one, x, we don't know what it is, we cannot edit to one. The same way as x minus two, we cannot edit x minus two, we cannot subtract them to one another because we don't know what x is. So we can solve the bracket, but we can distribute to remove the bracket. So we distribute three into the bracket in order for us to remove that bracket. So three times x is three x, three times positive one, it's plus three, plus four is equals to five, minus three times x is minus three x, minus three times minus two, it's positive six. Like terms together. When it's an equality sign, it doesn't matter whether you leave things on the left, sorry, whether you leave things on the left or you move them over to the right, but I always prefer things to move to the left. So let's move everything that has an x to the left and everything that does not have an x to the right. So three x, three x on this side, it's minus three x, when it comes this side, it will be plus three x is equals to five, plus six, and we move three and four to the other side, they are both minus, so we can say minus three, minus four. And you could have already simplified them here by saying three plus four is seven and move only minus seven to the other side, the same way five plus six is 11, you could just write the answer there, but I just like to complicate my life, so I'm gonna put it this way. Three x plus three x is six x, five plus six is 11, minus three is nine, 11 minus three is eight, minus four, you are not mad, it's four. And we just need to only be left with x on this side, so therefore we're going to divide this side by six and divide this side by six, six and six cancel out and you will be left with four over six on the other side, but there is a number that can go into four and also into six, so two goes two times into four and it goes three times into six and the answer will be two over three, which is option two. Easy, right? Easy, straightforward, easy, peasy. What happens when you get equations? And that looks like this, that looks so, as if like they are so confusing and they are not. As long as you remember the principle, if it's dividing, you need to multiply everything by that number. Whatever you do on the left, you must also do on the right. Now, the challenge here is you have two x over five minus x over three is equals to one over two. You cannot do the subtraction here or you can. It's up to you how there are many ways you can solve this. You can treat this as a fraction and solve this fraction on the left and once you have the answer for this fraction on the left, then you can simplify and find the x value. So let's do that. We're gonna do this two times two ways. Let's see if we get the same answer doing it two ways. So the first way is we treat this as a fraction. So we're going to solve two x over five minus x over three on the left. We're only going to treat the left hand side first. So two x over five minus x over three. Now I need to find the common denominator. Remember, it's addition and subtraction. We need to find the common denominator and the common denominator here is 15. Five goes how many times into 15? It goes five, 10, 15, three times two. Three times two is six x minus five. Three goes how many times into 15? It goes five times five times x is five x. And now what do we have? We will have six x minus five. It will be x over 15. Now, this is our left hand side. So if our left hand side is x over 15 equals to a half, then let's get rid of 15. We multiply the side by 15, multiply the side by 15 and therefore we will get 15 and 15 cancel out. You get x is equals to 15 over two. Remember, a fraction to simplify it to the lowest form until it's known that can be simplified is to convert a improper fraction to a mixed fraction. So two goes how many times into 15? It goes two, four, until you get to seven. So it goes seven times and the remainder will be one over two and the answer is option one. That's the first, the other way of doing this, right? The other way of doing this is we have two x over five minus x over three. So I'm gonna call this option one. And this is option two of answering the same question, right? x over three is equals to one over two. Now, because two over five is on the first one, what we need to do in order to get rid of two over five, we're going to multiply by two by the inverse of two over five on across. So it means we're going to say five over two multiply by two over five x minus, we also need to do it there. So five over two times x over three equals one over two times five over two. So because we want to get rid of the first two over five so that we are left with x. So that and that will cancel out. You will be left with x minus. And at the top here, we will have five x over six is equals to five over four, right? Because it's one times two, one times five and two times two is four. Now, the next is to get rid of the five over x. Now, because in order for us to get rid of this, it means we're going to multiply. It's going to be a non everlasting cycle of getting this multiplied by that multiplied by that again and all that. So we need to do that. So in order for us to get rid of five over x, we can treat this as a fraction because they are like times, right? Because five minus five x is a like time. So yeah, the common denominator is six because this is the same as x over one. So this will be six x minus five x. And therefore you will be left with x over six, five over four and we get rid of six by multiplying by six this side, multiply by six that side. And that will give us six and six will cancel out. You will be left with x, four goes two times into four. Two goes two times into four and it goes three times into three and your five times three is 15 over two, which is the same as seven, one over two. You can go the complex route or the simple route, it's up to you. What the fact remains is, the fact remains that you need to apply your mind when you solve equations. You need to think about everything you know with regards to fraction, with regards to powers, with regards to bot mass, with regards to expressions and all that you bring it and you answer your equations. The other way of answering equation is to making something a subject of the formula. And yeah, it is when, yeah, it is when you have multiple variables multiple unknown placeholders in your equation or your expression and we need to change the subject of the formula. So the subject of the formula, for example, with the exercises that we were doing, our subject of the formula here is x. When a variable is on its own, on the left hand side, we call that a subject of that formula or expression that it's on the right. So in order for us to change the subject of the formula, the same principle happens, whatever you do on the right, you do on the left, left, right. If you move things across, if it's dividing, you multiply, if it's multiplying, you divide. If it's a plus, it becomes subtraction. If it's subtracting, it becomes an addition. Those are the same principle, nothing changes. Done, going for, if we need to solve this formula. If S is equals to P times one plus RT, which is your simple interest formula for future value, future value of a simple interest. Make R the subject of the formula and therefore it means here they're asking you to make the rate, the subject of the formula. We want to calculate what the rate is, what the interest rate is, the simple interest rate will be. So in order for us to calculate that, you can see that R is inside the bracket and in the bracket, R is multiplying with a T. So in order for us to make R the subject of the formula, we need to remove certain things. So let's go, S is equals to P times one plus RT. The first thing that we need to get rid of is the P. So we divide the side by P, therefore it means we divide the side by P. P and P cancels out. Then you are left with one plus RT on the right hand side. The left hand side is over P. Now we need to get rid of one to the other side as well. This side we will be left with RT and when we move one to this side, it's S over P minus one. We can also change the equation. Let's rewrite it because it's an equal sign. So therefore it says the right is the same as the left. So we can also say RT is equals to S over P minus one. It will be the same thing because it's whether we write it on the left or on the right. There was still balance. Okay, so now we can solve for R. So solving for R, we need to get rid of T. Therefore it means we divide by T on one side, we divide by T on the other side. And T and T will cancel out. You are left with R and this will be S over P minus one over T. Now looking at this, you might think, oh yeah, there is no answer, yeah. How did she do it? Right, so we can create this S over P minus one divide by one, divide by T. You can rewrite it because this is a division, all right? Sorry. This is a division. So a division, we can write this as S over P minus one. And I'm gonna put it in the bracket and I'm gonna put the divide by, remember, now this is divide by T, right? What do we remember about fractions? Because if I convert this to a fraction, then I can keep the first S over P minus one and I can multiply and we change this to one over T, right? That's what we can do. We can keep, flip. So when we flip, it becomes one over T and we can then cross multiply with a T across because it's multiplying. So T will multiply with P, then you will have S over T minus one times one is one divide by, because this is the same as one divide by one will be over T and the answer will be option number four. So you need to apply your mind. I know that it's very confusing. You need to think outside of the box sometimes when you solve equations, especially when it comes to a equation like this, where you need to make something the subject of the formula as well and looking at the options, you need to apply your mind to say, am I at the end? If not, if there is no answer on there, it might be that you did the calculation but you did them wrong and you might find the answer. It doesn't mean that it's correct, right? But you need to think when you get two answers like this because we could have stopped at that point, at this point and looked for the answer here and would have penned and said, but there is no answer and usually sometimes in some exam papers they have option number five and they call it none of the above. And you might think that that is the answer and it's not because the answer is on there. You just needed to do a little bit of calculations as well or manipulations as well. While we're still talking about expression and equations, sometimes you might be given a sentence and you are asked to take a part or a paragraph or a sentence and convert it into a mathematical expression. So you need to also know those. Don't lose marks because it might be that you are losing four marks out of this and this is the easiest question that you can have, right? So don't lose marks by just not knowing how to answer questions like this. A father is three years older than five times the son's age. The father is three years older than five times the son's age. Suppose the father is ex-old, given expression in X for the son's age. And I said these things are very easy to answer, not necessarily, especially when it comes to literal equations like this. That's why you need to think long and hard on this type of questions because they can be as tricky as they are. So if we know that the father is ex, the father's age is ex, right? But we also know that the father is three years older than five times the son's age. So if the father's age is ex, therefore the father will be five times older than the son, but will be three years older. Therefore it means when it comes to the son, or not five, three, the son will be three years younger. So because the son is three years younger, the age will have to also be subtracted from the father's age. So the son is three years younger. So if the father's age is ex, the son will be three years younger because the father is three years older than five times the father's age. Ma'am, if you talk, are you quiet, ma'am, because I'm not hearing anything. I'm quiet because this is tricking me. It's getting me. So if we write it in this way, therefore it means the father is five times the son's age. So if the son's age is five, right? Five times the son's age, but the father is three years older than five times the son's age. So let's assume that the son's age is y. So the father will be three times that number. But we also know that he is older. I'm gonna put the three, right? And if this is the father, the father's age, right? So when we want to find the son's age, because that's what we want to find the son's age, but in relation to x, right? So it means we need to solve for y. So solving for y, we need to take three x will minus three onto the other side, which is correct as I did it. But then we need to divide by five. So we'll need to divide by five. So we're not going to multiply by five, because if we multiply this, we multiply in the father's age, that will be multiplied by five. So that is, yes, that's how you will answer it. So you need to go back and think about it. Because here it says the father is three years older than five times the son's age. So we don't know what the son's age is. I'm just gonna put the son's age as y. So five times the son's age, that's the father, but and also there is three year older, five times the son, so plus three. Suppose the father is x. So here is the father's age. The father's age is equals to five times the son's age plus three years, because it's all three year older than the son, unless, yes, yes, it's still right. So to find the son, we need to solve for y. So we will move three to the other side. That would mean x minus three. Therefore the son will be five times the son will be three year older than the father. But we also need just the son's age. So we need to get rid of the five and get ridding of the five. It means we divide by five. So probably the answer is option four. At some point I also thought that it might be option two. It might be that I'm doing this wrong. If it can be option four, if the following scenario happens this becomes five times y plus three. Then it will be option two, because then we're saying it's five times the son's age plus three, whichever one. And therefore it will be x divided by five and you will be left with y plus three on the other side. Therefore it will be minus three is equals to one. That will be the answer which will mean that option two is correct, not option four. But that is how you will read the questions. I don't know how to answer this one. It might be me confused. But yeah, that's how you will answer weighted questions. Now let's go back to another question. Sorry, there are no other expression questions. So those are ratios and the linear equations. We're not touching that. And let's find more exercises so that you are able to do a lot of exercises. So this exam paper doesn't look good at all. It's very blurry. Okay, so here is your question. A mother divides an amount of money amongst her children. Gahisa, Degeleri, and Tavo. Gahiso is, or is it Gahiso Gahisa now? Gahiso gets twice as much as his sister Degeleri. Degeleri gets 100 grand less than Tavo. Suppose Tavo gets x, then Gahisa gets x. How much does Gahiso get in terms of x? That's your exercise. Let's see if you are able to get that. I'm gonna give you five minutes to deal with that. And I will catch up, are you? And then we'll do it together. Are we winning? Ma'am, I'm not sure. I'm confused, but I think according to like, maybe how we worked at the previous one, then it could be option two. I'm not sure, but yeah, yeah. That is what I'm kind of getting. Let's read that question and then see if we are able to make sense of it. So we know that the money is divided that amongst the three children, you know? And Gahiso gets twice as much as Degeleri. So there is Gahiso. He gets twice as much as Degeleri, right? And we know that Degeleri, who is our G, gets 100 run less than table. This table minus 100. We are also told that table, let table become x. So we're going to change table to x minus 100, right? What we want to know is, or what they want to know is how much does gonna convert it back to D? D is equals to x minus 100. How much is Gahiso get in terms of x? Now, Gahiso gets twice as much as Degeleri. So therefore Gahiso will get two times Degeleri, which is D is x minus 100. And there is your answer. Yes, man, that was my first answer. And then I felt really confused and I thought, like, is that gonna be right? Because it's like, shouldn't we do it the same where we did the other one? But now I know I should just go with my understanding and not doubt myself. Yeah, sometimes it's easy. When you read the question and write out the fact that are given in terms of how you understand the question is being asked. It makes it easier to understand it. With this one, the thing that made the question more tricky is the mention of three years older and five times, where are we now? Three years older than five times the age of the sun. And they say the sun's age, but they say it represented in terms of the father's age of x, right? So it makes it a little bit difficult or really, we could have already also said, if the father's age is x, so we'll know that y will be, maybe we need to re-look at this as well. Y, which is the sun's age will be x plus three, but now because we know that it's not gonna be because we know that the sun's age is x plus three, then, or maybe we need to use father. Three years older than five times, then we can put the five in front or something like that, which makes it a little bit difficult to understand, right? Then with the first one, with the second one that we do it, because yeah, the second one, it's straightforward. They tell you that Cajizo gets twice as much as Degeleri. We are able to write it out as an equation. Degeleri gets less than Kabul. We are able to write there what Degeleri gets, because it gets less than what Kabul gets. And we are also told that Tavo is x, so we can replace Tavo with x on this equation, and then substitute back Degeleri's amount onto Cajizo, because then we know that Cajizo gets twice as much as Degeleri and Degeleri gets less than what Kabul gets, which is that equation, which is easy. So I hope in the exam they give you questions like this, which are straightforward. Okay. Thanks, ma'am. Your next exercise, simplify the expression as far as possible. This is x times x minus two, minus two times one, minus x squared, times x times x minus four x. Now, sometimes you get confused when you have an x after the brackets. What I can suggest you do is take the x and multiply it with the two first. So that will be x times x minus two, minus two x times one, minus x squared, minus four x, and then it will be easy for you to solve the equation. That's your chance to do that, and then we can talk about the answer. Are we winning? Yes, ma'am. Are we done? No. Are you guys there? Okay, so let's answer the question. Let's distribute. Sorry, ma'am, I'm here. Okay, so it's two x minus x, minus two x plus, oh, sorry, ma'am. Okay, I don't know where you are at right now. Let's distribute the x into the bracket, the first one. Yes, so it's x squared minus two x and then minus two x plus x two x cubed minus four x. And then if we take it according to putting it in order, like mathematical order, then it will be, we'll take the two x cubed and then we'll add all the, like there's only one x squared, so it will be plus x squared. And then we'll take all the other terms which is like terms, which is the minus two, minus two, so it's minus two x, minus two x, minus four x, which gives us minus eight x. No, ma'am, not that, yes. So the option one is the answer. Thank you. Your next question. Are we done? Yes, ma'am. Ma'am. Can I go ahead, ma'am? Yes, you can. Okay, so according to my understanding of the law of exponents, when it comes to square roots, you divide whatever the power is, for example, like b to the power of eight will become b to the power of four times by b to the power of eight and b and b stays b and then you add the exponents, which will be 12. So b12, so that will be option three. Yes, you can do that. You can also, because it's multiplication, you can also just multiply both of them. So it's b to the power of eight times b to the power of 16. And because they've got the same powers, you can just add, sorry, they've got the same basis. You can add the powers, so that will be 24. And we can take b to the power of 24 to the power of a half, which two goes one time and it goes 12 times and b's to the power of 12, which is the same we could have already from here, set b to the power of eight to the power over half times b to the power of 16 to the power over half. And we could have solved this two goes one time into two and it goes four times into eight times b goes eight times into 16. And therefore it will be b to the power of four plus eight, which is equals to b to the power of 12, whichever way you feel comfortable using the, of solving the expressions and equations, use that. So there are many ways you can answer the same question. The fact is in all those many ways, you still need to get to the same answer. Your next exercise, are we winning? Are we there? I can't believe that out of all the people, I only communicate with one person. Oh, guys, that's not fair. Are you guys here? Yes, we are. At least because it might be that you connected, but you are not here with us. You left the session. You are somewhere in the kitchen or somewhere watching TV and I'm just here alone with Michelle. Let's answer this question. Are you done? Ma'am, I'm not done. I'm lost. So I need a home. So it's almost similar to the one that we did previously. Remember, you need to first distribute the power of three into the whole bracket first because we have half A to the power of two, B to the power of three. So we just going to say two times one over two to the power of three by just distributing the three. A times, I can put the multiply by A to the power of two to the power of three times B to the power of three to the power of three. And we can solve this by saying two times one cubed is one. What is two to the power of three? It's eight because it's two times two times two. It's eight times A to the power two times three is six. And times B three times three is nine. Therefore, two goes one time into two and it goes four times into eight. One times one is one. One times four is four. A to the power of six, B to the power of nine, which is option two. Easy, right? What the following equation means make A the subject of the formula. When you are done, please let me know when you are done. So I don't have to ask you, are you done? I just want to hear how many people are done. Okay, I'll wait to hear a couple of more people. Done, babe. No, okay. Others, you can also type in the check. Okay. Oh, okay. So at least four people they look for it means a lot of you are done. Okay, who wants to try it? Okay, I'll try. Okay. First of all, I simply, I took what is on the left and I, I mean, what is on the right to the left? But I first simplified the brackets on my right. Okay, let's simplify the brackets. Okay, so the two times three is six. And then two times negative five A is minus 10 A. This will remain as is, okay? And then? Yes, and then I, the minus four A and then I took the 10 A and then it's going to be a positive plus 10 A and then it will be equal to six minus seven. And then I have six A on my left and then I have one. So I would divide by six, two, two, two. So it's minus one because it will take the sign of a bigger number, right? Yeah. So divide by six, divide by six, six and six, cancel. So it's number four. That would be nice. Okay. Hey. Yes, girl. Your next, solve the following equation. Five over nine X plus one over three is equals to five over six X. Same, right? When you are done, let me know. Ma'am, I'm struggling with this one. So others, I see you are all quiet as well. So let's get you unstuck so that we only left with 10 minutes before the end of the session. Okay, so this, you have X on the left, X on the right. So we put the items together. So five over nine X minus five over six X and we take the one without an X to the left, which will be minus one over three. What follows? We can solve the fraction. What is the common denominator between nine and six? Three. Mm-mm. Three, nine cannot divide into three. It's 18, right? Nine can go into 18 and six can go into 18, right? So the common denominator will be 18. Nine goes how many times into 18? It goes two times because nine plus nine is 18. Two times five, it's 10 X. Six goes how many times into 18? Six, 12, 18, three times. Three times five, it's 15 X is equals two minus one over three. Solve the top part, 10 X minus 15. It's minus five X over 18 is equals two minus one over three. Did you at least get to that stage? Now we need to get rid of minus five over 18 by multiplying by minus 18 over five. On that side, we also need to multiply by minus 18 over five on the other side. That will cancel out. You will be left with X is equals two and we can solve the fraction. What is the common denominator between five and three? It's 15, three. Oh, actually we don't even have to find the common denominator because it's multiplication, all right? This is multiplication. We can simplify. Three goes how many times into three? It goes one time. It goes how many times into 18? Three, six, nine, 12, 15, 18, six times. It goes six times. Negative times negative is positive. Therefore, the answer here will be six over five because this is an improper fraction. Five goes how many times into six? It goes one time. The remainder will be one over five. And the answer is option one. And that is how you will solve equations. I will do this one for you. You don't have to do anything. Simplify the following expression as far as possible. This expression, actually, the thing here is missing is the division. It's a division. So solving this equation, I'm hoping that is a division because if it's a subtraction, then it's something else. Let's start with it as a minus. Let's not convert it to a fraction and see if it works out. So we need to do what is inside the brackets first. So we have 18x squared divided by four y. And here at the top will be minus three x times eight. What is three times eight? It's 24, right? 24x over. Four times nine. Nine plus nine is 18. 18 plus 18 is 36. 36, y. And we need to find the common denominator between 36 and it will be a very big number. And I thought it cannot be. What will be that number that both of those two numbers can divide into and not leave a reminder? 36 times. It's how much? Is it 36? So 34 can go into 36, nine times. Yes, ma'am. So that will be 36. 36, y. That will be the common denominator. Four goes how many times into 36? It goes nine times. Nine times. Nine times 18x squared. So nine times 18. It's 162x squared minus 36 goes one time into 36. So that will be minus 24x. And when they look at things, this is how far it gets. There is nothing you can do about it. There is nothing you can. So it must be divided by probability. Yeah, so that is not the subtraction, it is a division. Okay, so if that is a division, then we will have 18x squared over four, y. Divide. What did we say this was? 24x. 24x over 36, y. 36, y. Now keep, change, flip, right? Yes. 18x squared over four, y. Change the sign to a multiplication. Flip the numerator and the denominator. And now we can simplify. So what are the numbers that can divide into each one of them? So can six go into 18 and can it go into 24? Yes, ma'am. Yes, and okay, so six goes into 18 three times, right? And it goes into 24, four times. You want to have it right? Yes, and is there a number that can go into 36 and also go into four? So four can divide into 36. Yes, it goes nine times. So four goes one time into, and it goes nine times and y and y cancels out. But also we didn't cancel out because it's x squared. So yeah, the x will cancel the one x there. We will be left with only one of them. Is the same thing as x squared divide by x because it's division, we say x two minus one because there is the power of one, which is the same thing that we have there. And therefore the answer will be x to the power, no, x to the power of one, which is the same as x. So yeah, we will be left with only x. So three times nine, what is three times nine? Or is there any other number that can divide into those? Nope, three times nine is 27, 27x. And one times four is four. And the answer is option two. And that concludes our session for today. So you can go and do additional other exercises. So please, you can see that the majority of your first questions are more about expression and equations and all that. You cannot lose out on this max. They are very important. So eight questions already in one example includes expression and equation. So you cannot, others we dealt with. So also practice, questions like this, changing the subject of the formulas and all that. On the previous other papers, you can see that question one is about expression. So you just need to make sure that you know how to solve them about expression, about powers and expression. So you cannot miss out on this max. And that is another expression. And especially those ones where they are written in words, so you can see that this we will do next week. Next week we'll do areas. So please go and practice. On this other question paper, they do have a couple of them. Those who attended the session last week have access to these three past exam paper that I shared with you. Otherwise, I will see you next week. Are there any questions, comments, theories before we close this session? Any comments, questions? If there are no comments or questions, then I will see you next week. Same place, same time. When we deal with measurements. Those who are doing QMI, I will see you later today when we deal with questions relating to data handling at 12 o'clock. Other than that, have a lovely Saturday and if we can't enjoy yourself. See you next week. Thank you very much. Thank you. Thank you very much. Thank you. Bye. Thank you. Bye.