 And so it's already time for the third lecture. We're going to look at powers, and those powers are also going to include roots, and then finally add some logarithms. Now we use logarithms as well as powers all the time in data science, so this is a very important lecture. And so distribution. How we distribute numbers, constants, variables, all of these interesting things. And I've mentioned this is so important. We use distribution all the time. So I'm going to use a couple of variables here just so that we can come up with some kind of definition. So we're going to say X times A plus B plus C. And what we mean by that is that's a variable. These are variables. Now usually these refer to whole numbers, constants. This is usually real number, but to see them all as variables. And what we're going to do is we're going to do this multiplication. We're going to distribute this X across this expression here. And so that we have, there's a multiplication symbol there. You can put multiply there. You put a dot there. But usually we put nothing. Just getting used to the way that we write things in mathematics. So this will be X times A. Another thing that you can learn is the order in which we put variables. We usually tend to keep these in alphabetical order. So X times A, remember, multiplication commutes. So X times A equals A times X. It doesn't really matter what I do. So I'm going to say AX. And then I'm going to do plus BX. And then I'm going to do plus CX. And that means A times X plus B times X plus C times X. I'm distributing this X into all under the solve expression which is three terms, a bunch of terms. And so let's make that the way that we're going to define how we do distribution. So that's going to be our blueprint there, our definition there for how we do distribution. Now let's just work our way through so many examples, as many as we can in the shortest amount of time and also remember what we've learned before. So this is a good exercise when we go through these examples just to bring in the previous lectures. So let's do our very first example. And in our very first example, we're going to see, let's make this X the number three and I'm going to say three times A plus B plus C. And I'm just going to take this three and I'm going to distribute it into this expression multiplying by each of the terms. So that's going to be three times A is three A plus three times B, we just write as three B. We don't put multiplication or dot in between. Although you can, there's nothing wrong with it. But this is just the way that we normally write mathematics and three times C is plus three C. So in this instance, my X was just equal to the number three. Next example, we're going to just do so many of these. Let's do two times, let's have four X, four times X. Let's do plus three times Y, let's do minus six. So how would we go about that? Remember this is this multiplication, multiplication, each of those. So two times four times X. Now let's just think it through. It's actually, it's two times four times X. And we can use the distributive property here. Two times four is eight times X. Well, this is going to be eight X. And so we have two times four X is eight times X. And so let's do the next one, two times three. So let's do that two times three times Y. Well, two times three is six Y, six Y. Once again, it's two times three times Y. And I'm making use of my distributive property. So I'm going to multiply those first. That will be six times Y and that's the six Y. So I've got my six Y. And lastly, two times negative six. So I'm saying two times negative six here. And remember, I'm just writing two times negative six. And there's a little one there. And this is back to what we've learned many lectures ago. And that's just going to give me a negative 12. So that means negative 12. Next example, let's do negative one. I'm going to have negative one times two X squared. Let's do minus three X. Let's do plus four. And that's not the way that we usually write it. We just drop that one altogether and we would just write negative two X squared minus three X plus four. And when we do distribute that negative in, it's actually this negative one that we are distributing in. So we're thinking negative one times two X. So that's a negative one times two times X to the power two. And I know those look the same, but I can do this first. Negative two times one is negative two X squared. And so I would have, let's write that there, just distribute the negative in. That's negative, let's put that there, negative two X squared. Now what happens if I have negative one and I multiply that by negative one? Well, remember a negative times a negative is a positive. So here negative one times negative three, that's positive three and they're still the X. And then negative one times four, well, that's negative four. So in the end, it's actually not that difficult. Next example, let's do one where we actually have a variable upfront. Let's do four X and I'm going to multiply that by X minus, let's do X minus two. And then I'm gonna have another negative and let's have five X plus three. Can we do this? Now there's a lot going on here. Let's tackle this first term, there we have a negative, so we have a second term in our expression. I have four times X times X. And if I had X times X, remember that's to the power one, that's to the power one, my bases are the same, so I can say X one plus one and that equals X squared. So I know four times X times X, well, that's gonna be four X squared. And then I have four times X times negative two and I can use my commutative property. So let's put the negative two up front. So negative two times four times X. I can do my distributive property and do this first, negative two times four, that's negative eight times X, that's negative eight X. So four X times negative two, well, that's just negative eight X. Now let's look at the second part. Let's just look at I've got negative five X plus three. That's the same as putting a little one there, remember? So negative one times five X, that's gonna be negative five X, negative one times three is negative three. So this becomes negative five X, negative three. Now I'm not done yet, I can simplify this. Look at this, I have negative eight X and I have negative five X. Now let's just think about it. And again, this is just back at school. Imagine I had two apples, two apples and I add to that another three apples. Well, what do I have? I have two plus three, that's five apples. And so you can think of those apples as a variable X and as a variable X, so two X plus three X, well, that's just five X. And same goes for here. Minus eight X, minus five X, negative eight, subtract another five, that leaves me with negative 13. So here we're gonna have four X squared, minus 13 X, minus three. So a very important thing for us to remember. If my variable here is exactly the same, I can think about that X as anything. So I might as well write apples there. And indeed, if I have two apples and I add to that another three apples, think about it, I've got two apples, another three apples, altogether there are five apples. So if this was an X and that was an X, I would have two X plus three X would just be five X. Just a single X there. I'm not squaring those X's, I'm adding those two. It's two apples and three apples. I've got negative eight apples and I take away another five, negative eight apples, take away another five apples, that leaves me with negative 13 apples. Great. So we've learned about this distribution, we've learned about the negative one and we've learned about simplifying when I have exactly the same variable. Thinking of them as nothing other than apples. Next great example. Let's do the variable a and I'm going to do, let's do a to the power four plus two a to the power two. Let's make it plus three. So what's going on here? I've got a times a to the power four, a times a to the power four. I remember that's a one. I've got the same base. I've got multiplication. So that's going to be a one plus four, a raised to the power one plus four. I add those two powers and that leaves me a to the power five. So a times a to the power four, you learn to do these very quickly in your head. That's going to be a to the power five. Let's do the next one in our head. There's a times two times a squared. Now, commutative property. I can do the multiplication in any order. Let me do the two first. I'm going to do the two first. And now it's a times a to the power three. There's a little one there. One plus three is four. Now, let's just remind ourselves, did I say this was squared or did I say cubed? So definitely, I think this for me is a two. So one plus two is three. So that'll be a to the power three. And a times three, commutative, is just the same as three times a. So I can just like three a. So we like to keep these in alphabetical order as we did there, but we also like to put numbers first and then variables. So a great little example. Let's do another one. Let's have a to the power four. And I'm going to multiply that by a to the power three plus let's make a to the power one over four. So I have a radical, my radicand, my index. What's going to happen here? Well, I'm still going to do distribution. Let me do it here at the bottom. So let's do a to the power four times a to the power three. The bases are the same. So that's going to be a, and I just add these powers and that equals a to the power seven. So I know this is going to give me a to the power seven, but now I've got a to the power four and a to the power a quarter. So I've got a to the power four times a to the power a quarter. Bases are the same. And I know I can just add these four plus a quarter. Now how do I do that? And I think that's a brilliant example of just going back to what we've learned before. So let's just put that way up here. Four plus one over four. Now I can rewrite this four as four over one plus one over four. Four divided by one, anything divided by one just stays that thing. What we want is the least common multiple and what's the least common multiple between one and four? Well hopefully you can very quickly see that that is just four. So I can multiply this by one and I'm just going to rewrite that one as four over four. I have changed nothing if I just multiply any number by one. How did I know this has to be a four because the least common multiple between one and four is four? Do you want me to prove that? Let's have a look at that. Remember I'm going to break these into my prime numbers. So there's the number one and the number four. Now one is not a prime number. There's nothing I can do with one. It is just one. Now four can be divided by the first prime, which is two, which leaves me with two. Two can be divided by the first prime, which is two. There's another two. Two times two is four and now I'm just going to bring them all down. In that column there's a two. In that column there's a two and two times two is four. So the least common multiple of one and four is four. Now I take my four and I multiply by one, but I'm not going to write one as such. I'm going to write it as four over four. Now I do remember that I have two fractions and I'm multiplying them, which means I can multiply the two denominators. Four times four is 16. Multiply the two denominators. Sorry, that's the two numerators. The two denominators, that is four, plus one over four. And now I have the same denominators. Now I can actually do addition. So 16 plus one, that's 17 over four. I have this common denominator. So eight to the power four times eight to the power quarter actually is going to be plus. There's a plus eight to the power 17 over four. That is a difficult one, but it allowed us to remember that how do I add two fractions if they don't have the same denominator? Well, I've got to make their denominator the same and what I'm always after is the least common multiple denominator. And for me to use the least common denominator, I have to remember to factorize my number into a product of primes. And so we did that right there. Now let's do what I think is a very complicated example. So let's do something like this five times eight to the power, let's make that negative three. Let's do B to the power negative two. Now we do remember how to deal with those. Let's multiply that by two times, let's make it a, B to the power three, which we call B cubed. And let's subtract from that, say three. Let's do a squared, B squared. A to the power two, B to the power two. Now that looks horrendously complicated, but I've still got one term here which I'm going to distribute into this expression. So this term times that term and this term times this term with a negative out front. So let's do that and we do remember that when we deal with multiplication that we do have the commutative property. So the order of multiplication doesn't matter. But now I'm going to do these two terms. So let's just put a line under these. So we just see what we're talking about. So here's my full term here up front instead of a single three or A or X. I now have these terms that are slightly more complex. So there are the terms that I'm dealing with. So I'm going to multiply this term by this term and I'm going to multiply this term by that term. And I do remember that there's a negative out front there. So let's look at these. As I say, the order doesn't matter. So let's do the five and the two. So five times two. Now I've got A to the power negative three. Let's say A to the power negative three. And let's put the other A there. That's A. And remember, that's just to the power one. Here's my B to the power negative two and there's my B to the power three. Now for the next term, I've got five and a negative three. So that's negative. Let's do five times three. I've got my A to the power negative three from the side. I've got my A squared on the side. I've got my B to the power negative two here and I've got my B squared from there. And now I just have to remember I've got the same base, same base, same base, same base. And now I can just add these exponents or these powers. Five times two is gonna equal 10. Let's do A, negative three plus one leaves me with negative two. I have my B, negative two plus three leaves me with a one. Negative five times three, well, that leaves me with negative 15. Negative three, there's an A and A. So negative three plus two is A to the power negative one and negative two plus two is zero. So that leaves me with B to the power zero. And let's really write this. Let's clean that up a bit. So that's gonna be A, 10 times A to the power negative two. B to the power one, we can just leave as B minus 15. I've got A to the power negative one and anything other than zero to the power zero is just one, so I'm gonna leave that out. Can I simplify even more? Yes, I do remember what happens when I write A to the power negative two. Well, that's just gonna be one over A squared. And similar there, A to the power negative one is just one over A. So if I really want to rewrite this, I can say 10 B times B over A squared minus 15 over A. Great. Now, can I get at least common multiple between those two? Yes, I can, but I think this is a neat way to leave the solution of this distribution problem. So one of the things that you might see in textbook if we have to do this, we might call this an example. We say expand the following. And now I've expanded it. I've multiplied these two terms. I've multiplied those two terms and I've done some expansion. Now, we've had those negative exponents. Let's do something even more fun. Let's do the following. The square root of x times y, and I'm going to multiply that by another square root of x times y. And I'm going to say plus the square root of y. Now, there are so many ways that we can deal with it. There's just absolutely so many ways that we can deal with this. And we can deal with it in a very rigorous way or we can just have some fun with it. So let's start just with a fun way. I've still got my terms. And again, what I'm going to do is I'm just going to put all my terms together like this. There's one term, there's one term and I've got the positive there. There's my other term. So now we're going to multiply those two and we're going to multiply those two. So in actual fact, we have the square root of xy times the square root of xy plus I've got the square root of xy here and multiply it by the square root of y. Now we have to remember this property which states that if I have say the square root of a and I multiply that by the square root of b, now remember we have the fact that the index is the same and the index goes there. We don't write that, but that's two for square root. And that means I can just combine these a times b. So if I had them separate, I can combine them. So the same is going to happen here but let's just do it more rigorously. Remember when we had this idea of the square root of c times the square root of c, irrespective of whether that c was positive or negative, if it was negative, I couldn't do that because I cannot take the square root of a negative number. Taking the square root, let's put it out here. If I have four to the power half, now remember I'm going to write that as four to the power one and my little index goes there. So I'm asking what is the square root of four? Well that is going to be plus or minus two. Why? Because if I take plus or minus two and I square them, so two times two is four. There's my four, negative two times negative two is also positive four. So it's both positive and negative. But irrespective of whether that is a positive or negative two, if I square it, I'm always going to get a positive number. And hence I cannot take the square root of a negative number as far as getting a solution where the solution's actually a real number. Can I do it with a complex? The complex numbers, of course, I can take square roots of negatives, but that's not what we're dealing with. We're dealing with real numbers. Remember that we said that that would be the square root of c times c, which was equal to the square root of c squared and that was equal to the absolute value of c. We just want the fact that we really do need just the positive version of these, reflecting the fact that we want to take the square root of positive numbers. So what we have here is the square root of x times y times x times y, leaving us with x squared, y squared. There's two x's in there and two y's. One plus one and one plus one leaving me with the twos. So we have the square root, there's a single x and there's a y squared in there. Now I want to do all of this in reverse. If I had the square root of a times the square root of b, I had the square root of a times b, but I can also work backwards. So I can do these in reverse. So I can say, well, that's going to be an x squared times the square root of y squared. And here I'm going to have the square root of x and the square root of y squared. Now I do remember the fact that this can be written as x to the power two to the power a half. There's my x squared, there's a little two there that becomes x squared to the power half. This becomes y squared to the power a half. That stays the square root of x and this one becomes y squared to the power a half. And I do remember if I have something and I take it to the power, so that's something with the power to the power that I multiply these two. What's two times a half now? That's just equal to one. So I'm left with x to the power one. I'm left with y here to the power one. Plus I'm left with the square root of x there. In here I'm left with y to the power one. So in the end I have xy plus the square root of x times y. So that's quite a long problem for us to deal with. I also just want to make sure that we do have this idea here that if we have x squared, that will be the same there as c squared. It's just available, it's just available. That I must consider here that I only have the absolute values and that strictly speaking these should be absolute values. Let's do an even more complicated example. Let's do the square root and remember I usually don't put the two there but let's put it here for now. Let's do xy, let's do cubed. And I'm going to multiply that again by the square root. Let's make this x cubed y and I'm going to have another, let's do that. Let's have another square root. Let's have xy to the seventh. So that is quite complicated. Again, I'm just going to have my terms and I'm just going to multiply these. So there's my term up front. Here's my term and there's my second term. So again I'm going to multiply those two and I'm going to multiply those two. Let's see where we get. And I'm going to go straight off the bat by remembering that this is a square root, that's a square root, there's also two there that I can just combine all of them in one. So I'm going to have an x times an x to the power three. There's a little one there. So one plus three is four. So I'm going to have x to the power four. And I'm going to have y and there's a little one there as well. Three plus one is four. So I'm going to have y to the power four. And there I have a positive and I'm going to put everything under the same roof again. There's a one, there's a one. So that leaves me with x squared and there's a three and there's a seven. Both are y's. So that leaves me with y to the power 10. So can I simplify these? Yes I can and there's a variety of ways that I can go about that. Remember we can break them apart. Let's do that. I'm going to have x to the power four, that little square root. And here's another little square root. Plus here's another little square root. And here's another little square root. So I can break them all apart because I do remember the square root of A times the square root of B that is the square root of A times B. And I can also work in reverse which is just what I've done here. I've got AB and I'm just making them two separate. And I do remember that I can rewrite this in another way. That's x to the power four to the power a half. Remember if I have the square root of A that's going to equal the square root or at least A to the power one half. Those are exactly the same thing. And here I have y to the power four and I also have a half there. So I'm turning the square root into a half. Plus here I have x squared. I'm going to do that again. And here I have y to the power 10 and I'm going to take that to the power one half. And now it becomes easy because four times a half is just two. So here I have x squared. Here I have y squared. Plus here I have x squared. Or at least I should say two times a half is just one. There's just one left there. And here I have y 10 times a half is five. So I'm going to have x squared, y squared plus x y to the power five. Now we're going to make things even more complicated. Instead of just a single term in the front, let's do something else. I'm going to have three x and I'm going to have plus four. And then I'm going to multiply that by, let's do two x, let's do negative one. Now, how about this? Look at this. I have two terms here in this expression. That's before I only had single terms. Now I can do exactly the same thing. I can still say that this is my little expression up front. It is now two terms, but let's keep it as such. Here's another one and there's another one. And I can still do the exact same thing. Later on we'll see there's an easier way to do this. Not an easier way, but the way that is commonly taught. But I still have something in the front of a two term or multiple term expression. And I'm still going to do this whole thing. What I do remember though is that I have the commutative property. So if I have three x plus four and I multiply that by two x, it'll be the same as two x multiplied by three x plus four. Let's do that. Let's have two x and I'm multiplying that by this first term, which was three x plus four. And now I have this term times a negative one, but it can commute. So I can also do negative one times this term up front or the expression up front. So that's going to be a negative one and I'm multiplying that by three x plus four. And now, now I'm just where I was before. And now I have a single term out here. I still have my two terms here. I have now this negative one out here, which is actually a term. And then these two terms out here. So I'm still going to do this multiplication and then this multiplication. Nothing before. Now I have two x times three x. So I've got two times x times three times x. I can do the commutative property. So I can do two times three times x times x. The bases are the same. There's a little one and a little one. One plus one is two. So this will be six x squared. So two x times three x gives me six x squared. And two x times four, by now you should know that this two times four is eight x. The negative is going to distribute in. That's going to be negative three x. And that's going to be negative four. And again, I have this eight apples. Take away three apples. I'm left with five apples. So this is going to be six x squared plus five x minus four. Six x squared plus five x minus four. But as I mentioned, there's another way that we usually write this. And we explain it at least. There's my three x plus four. And there's my two x minus one. And what we can do is to see all of these as individual terms. There's a term together with a positive four. There's a term, there's a term, and there's a term. And now what we can do is just link everything up. Now look at these. This one was linked with that one. This one was linked with that one. They're all linked. Now I'm going to do the same here. So this one is linked with this one. And it's linked with that one. But let's also do this at the top. Let's just put our terms out here at the top as well just to show you what's going to happen. Exactly the same terms that I have. So I had the three x that I distribute over this. But I'm also going to distribute this four as well. So that's what you're going to see. And so let's do this. The three x times the two x, we still remember that to be six x squared. I'm going to take the three x minus one. Three x times negative one. That leaves me with negative three x. I'm going to have the four times two x. That leaves me with a positive eight x. And I'm going to have the positive four times the negative one. That leaves me with negative four. Now I still have eight apples. Take away three apples, leaving me with five apples. So this is still going to be six x squared plus five x minus four. Exactly the same solution as we had before. So whether you stick with our definition of distribution that we had in the beginning, that it's this whole term that we're distributing to each of these terms here, you can also think of it as breaking everything apart. So I've got two terms here and two terms here. And so I'm going to take the three x and distribute that over those two terms. And I'm going to take the positive four and I'm going to distribute that over those terms as well. Now we're having so much fun we can just keep on going, going, going. So let's make things even more complicated. Let's have two terms here, two x minus four. So that's two times x minus four. And I'm going to multiply that by x squared. Let's do plus x and let's do negative one. Now again, what we could do is we could see this as a single expression. And we're going to multiply this whole expression by x squared, this whole expression by x and this whole expression by negative one. Or we can see them as individual terms. And so we're going to do the following. I'm going to take the two x multiplied by the x squared. I'm going to take the two x multiplied by the x. I'm going to take the two x and I'm going to multiply it by the negative one. Then I'm going to take the negative four, the whole of the second term. I'm going to multiply it by x squared, negative four times x and negative four times negative one. So I'm linking up each one of these with each one of those. This one with all of those as well. So let's do this this way around. So two x, so I've got two times x times x to the power two. There's a little one there. The bases are the same. I can add those two, one plus two is three. So that's going to be two x cubed or two x to the power three. So two x to the power three. Now I'm going to take my two x multiplied by positive x. So that's two times x times x. There's a little one, one plus one is two because these two bases are the same. There's going to be two x squared or two x to the power two. So that's two x squared and two x plus negative one, that just leaves me with negative two x. And now I'm going to carry on. Now I'm going to do the second term, the negative four, and I'm going to distribute that through each of these terms in the expression as well. So negative four times x squared leaves me with negative four x squared. Negative four plus x leaves me with negative four x. And negative four plus a negative one, negative times the negative is a positive and four times one is four leaving me with plus four. Now I just have to group the terms. There's only one x cubed here. So let's leave that alone, two x cubed. But I have two x squared, negative four x squared. Once again, I can think of x squared as being apples and apples. So I'm saying two apples, take away four apples, leaves me with negative two apples. So there's negative two and there's my apples which are x squared at the moment. Same thing here, I've got negative two apples. Take away another four apples, negative two times negative two, negative four. So that leaves me with negative six apples or just x. Now I call this apples as well. Strictly speaking, I should have been talking about apple squared, but this is just one single thing. If I take three and I squared it, I get nine and nine is just a single thing. So I'm just seeing x squared as a single thing. And so I've got two things and I'm taking away four things, leaving me with negative two things. I've got negative two things and now I don't, you know, that x, if x is nine, this is nine, whatever the number is, I've got negative two of those things, negative four of those things leaves me with negative six of those things. And at the end, I've got the positive four left. So two x cubed minus two x squared minus six x plus four. That was a very fun example. Now I want to do one more for you. Let's do, let's do a plus b and we're going to square a plus b. Now what does that mean? Well, square means you just take whatever and you just multiply by itself so many times. There's my a plus b. And once again, if this was three and this was two, two plus three is five. So I'm taking five squared. I'm taking five, multiplying it by itself twice. I'm taking a plus b, multiplying it by itself twice. Once again, you can think of this as a single unit, multiplying this by a, this by b, or you can do this individually. And let's stick with this, doing this individually. So it's a times a. And remember, a times a, there's a one, there's a one, bases are the same. It's going to leave me with one plus one, that's a squared. One plus one as far as the powers are concerned. So this is a squared. I have a times b, that's a b. Now the second term, b times a is b a, but this commutativity and we keep with the order of the alphabet. So that's another a b. And then b plus b times b, I should say is b squared. Now I've got a b plus a b. I can think of a b as apples again. So one apple plus one apple gives me two apples. So that's going to be a squared plus twice a b plus b squared. That's an interesting one. Now I want to do a plus b cubed. So I'm going to have a plus b times a plus b. Oh, times a plus b, what now? You know, I would have had to draw so many of these, but I do remember the distributive property at least. And so I've already done this bit. That is a plus b a plus b. So I can rewrite this at least here as a squared plus two a b plus b squared. And then I'm still left with this a plus b right up front. So we will just do it piece by piece. And now I can do my a distributed throughout and my b distributed throughout. And a times a squared, I'm going to do this much quicker now. There's a one and a two that leave me with a three. Plus I've got a times two a b, that's two a squared b. I have a times b squared, that leaves me with a b squared. I have b times a squared, I'm just going to swap that around so that's a squared b. I have b times two times a times b, so that leaves me with two a b squared. And lastly I have b times b squared, that leaves me with b cubed. And now just similar terms. There's an a to the power three. And so I have a two a squared b and there's an a squared b there as well. A squared b, a squared b. Two of them and one of them makes three of them. Three a squared b. And there's an a b squared and a two a b squared. I have something plus another two of them makes it three a b squared. And finally I just have the b cubed at the end. So we've just seen what it was like to expand for instance a plus b squared. What I want to introduce you in this bonus section is this Pascal's Triangle. Pascal's Triangle and that's going to help us expand certain expressions. Let's have a look at this triangle. The triangle starts with a one and then it builds down as a pyramid. So I'll have one on the outer sides always. And now let's do one more level. There's a one, there's a one. But now I'm going to look at one plus one and that gives me two right here in the middle. If I look at the one and the two that gives me a three. If I look at two plus one that gives me three and I add ones to the end. One plus three is four, three plus three is six, three plus one is four and I put one at the end. Let's do it one more time. One plus four is five, four plus six is 10, six plus four is 10, four plus one is five and I put one in the ends. And this is what we talk about when we are going to do something like x plus y all squared or a plus b all squared. And so let's have a look if I just have on this level right here I have a plus b but I'm going to take that to the power of one. That's just going to leave me with a plus b. And we see there's a one as a coefficient. We'll later on we'll have a close look at coefficients and there's another one and that corresponds to the one and the one there. And so right up here I suppose you can say this will be a plus b to the power of zero and anything that's not zero to the power of zero is equal to one and that's the single one that we see there. Let's have a look at what we do when we get to this level a, there we go a plus b, let's square that and we get a squared plus two a b plus b squared. And there we go the coefficients. There's a one, there's a two and there's a one. Let's do the next one now we've done these so you know what they should be the squared and the cube. Well let's just try to use what we have here. We're definitely going to have a to the power of three. So there's definitely going to be an a cubed and then there's going to be a three and a three and another one right at the end but let's make this one b cubed right at the end because we know right at the end we are going to have b times b times b. And all we have to do now is to go down in powers of three and go up in powers of two. So this would be b to the power of zero which is this one, that's fine. This one's going to be an a squared. This one's going to be an a and then there's going to be an a to the power of zero which is this one. So let's work backwards with a b. So I'm going b cubed, I'm going to be squared. I'm going to go to b to the power of one and b to the power of zero which is just one. And there's the coefficients one, three, three, one. And we just have the decreasing powers of a cubed, a squared, a and a to the power of zero and then in reverse we're going to go b cubed, b squared, b and b to the power of zero which is just one. Now let's try and do this very last one. Now we're going to start with a one and that's going to be a to the power. Now this is the fourth level. So let's put it there, a plus b to the power of five. So let's have a look at then. There's definitely going to be an a to the power of five plus now we're gonna go down in powers of a. So a to the power of four, there's going to be an a to the power of three. There's definitely going to be an a to the power two. There's going to be an a and then a to the power of zero which is just one. And here by the a to the power of zero we're going to start with b in reverse order, b to the power of five, b to the power of four, b to the power of three, b to the power of two, B to the power one, and B to the power zero, so there'll be nothing there. And all that is left is for me to put in these coefficients. So there's B of one, there'll be a five, there'll be a 10, there'll be a 10, there'll be a five, and there'll be a one. And that's what we are left with. A to the power five plus five A to the power four B plus 10 A cubed B squared plus 10 A squared B cubed plus five A B to the power four plus B to the power five. And so if you use Pascal's Triangle, it's very easy for us to expand something like X plus Y or A plus B, whatever these two variables are.