Math Olympiad|Japanese|Algebra.

Let me show you the easiest way to solve this type of question. We are asked to solve for a to the power of 11 plus b to the power of 11. 11. This is what we need. We need a to the power of 5 plus b to the fifth power multiplied by a to the sixth power plus b to the sixth power to get this Why do I say that, because of the law of indices or exponents, we are told that a to the power of 5 multiplied by a to the power of 6; that is, if you have the same base in the law of exponents, you are going to pick one base once they are multiplying, then you add the power; you can see that we will get what a to the power of 11, which is this; similarly, if you multiply this by this, we get What a B to the power of 11. So everything we are going to do now, we are going to concentrate. on how to get a to the power of 5 plus b to the power of 5 and a to the power of 6 plus b to the Power of 6, so let's do this together. First, we need A and B. Yes, you are going to see why, and to get a b We are going to square this first equation, so let's call this Equation One and this equation two, so if I square equation one, all right, I'm going to have what a plus b squared, okay, is equal to one squared now this is what algebra identity what does a plus b squared in algebra identity stand for what it is equal to is what a squared plus b squared plus two a b all right is equal to one squared we give you one; now record that a squared plus b squared equals two, so that means you're going to replace this with what two plus two equals one now record I told you that we need a b, so that is exactly why we had to square this; now we are going to do what to find To get our A B, we need to bring out two A B, right? All right. Now, two A B is equal to one, then this. Plus three times three cross becomes minus two, so this implies that two a b is equal to minus one. Alright, now to get a b, we need to divide both sides of this equation by two, so when we do that We are going to have that two; we can see two, right? This is what we have: two A B is equal to. minus one over two, so let's divide this side by. Sorry, what we have is two a b, that is, in case you cannot see here what we have is two a b is equal to minus one, then we are going to divide both sides by what two and when we do that we have that our a b is equal to minus one over two this is a very Important equation, so let's call it Equation Three. Now we have found our a and b, recall that I said that We need what we need: a to the power of five, right? How do we get a to the power of five if we have a squared to get a to the fifth power we need a to the third power, yes, because when we multiply a to the power of two by a to the power of three according to the law of indices, we are going to get what a to the power of five, so that means we need to cube this to get a cubed since we already have a squared, so equation one cubed is going to give us what a plus b raised to the power of three is equal to one raised to the power of three in algebra is the same thing as a to the power of three plus b to the power of three plus three a squared b plus three b squared a, all right, is equal to one because one to the The power of three, we still give you one; I want to stand there because it's better for me this way. All right, but we have a to the power of three plus b to the power of three; that is what we are looking for. for right plus now we can factorize this as what three a b multiplied by now three a squared b divided by this we get what one a remaining plus three b squared a divided by this we get what one b remaining is equal to one, so we have a to the power of three plus b to the power of three plus three, recall that our a b is minus one over two, right, minus one over two (a + b) is equal to one is equal to one, yes, according to this equation, so we have a to the power of three plus b to the power of three plus so this is going to be minus three over two, this is going to be minus three over two, which is equal to one. All right, permit me to Draw this line; now we can transfer it to the other side of the equation, so we are going to have that a to the power of three plus b to the power of three is equal to one now this minus we cross to become plus three over two, okay, so we have a to the power of three plus b to the power of three is equal to if we will if we use I want to use a shortcut to solve this so if we use Butterfly method: this time we are going to have this, which will give us the denominator, which is two. Now we are going to cross multiply: one times two gives us two, plus three times one gives us three is equal to five over two; now we have gotten a to the power a cubed plus b cubed. Now let's call this Equation Four. Recall why we are looking. for this, because we want to get a to the power of five, right now, perfect, we have a to the power a cubed plus b cubed, so let's multiply equation four by equation two to get a to the fifth power; so we have equation two multiplied by equation four, so it's going to be a squared plus b squared, right, a squared plus b squared multiplied by a to the power of three b cubed plus b to the power of three equals two times five over two times two multiplied by five over two so if we expand this we are going to have a to the power of two times eight to the power of 3 is going to be a to the power of 2 plus 3, which is 5, all right, plus a to the power of 2. multiplied by b cubed gives us a squared b cubed plus b squared times this, we get b squared times a cubed plus b squared times b to the power of 3 gives us b to the power of 5, right? is equal to this; we divide this, and we have 5 remaining, so we now have a to the power of 5. plus b to the power of 5, which is what we are looking for. Okay, now look at this; we have a squared b squared is common here, so let's factorize; when this is divided by this, we have b remaining so b plus when this is divided by this we have a remaining is equal to 5 now let's substitute so we have a to the power of 5 plus b to the power of 5 plus now this same thing as What is a times b squared, right, multiplied by b plus a, equal to 5? Okay, let's move to this side of the Okay, the equation is correct, but before that, we need a to the power of 3 plus b to the power of 3. So let me write it somewhere: a to the power of 3 plus b to the power of 3 is equal to what did We get that it is equal to 5 over 2. Okay, now I can clean this up. Now I can clean it up. Okay, so let's continue now this is what we have a to the power of 5 plus b to the power of 5 so we have a to the power of 5 plus b to the power of 5 oh let me turn like this so that my writing will be okay so we have a to the power of 5 plus b to the power of 5 plus now recall that our a b is minus 1 over 2, yes, so we are going to have minus 1 over 2 squared, okay, bracket b plus a is 1, is equal to 5 so we have a to the power of 5 plus b to the power of 5 minus 1 squared we give you 1 and 2 squared, we give you 4; this times 1, we still give you 1 over 4 is equal to 5, so we have that a to the fifth power of 5 plus the fifth power of b, which is what we are looking for, is equal to 5 minus 1 over All right, so a to the power of 5 plus b to the power of 5 is equal to, let's use our shortcut as Let's cross multiply: 5 times 4 is 20, minus 1 times 1 is 1, so you have a to the power of 5. plus b to the power of 5 is 19 over 4, perfect. So this is our equation 5. Now we have a to the power of 5 plus b to the power of 5. Recall that all we have been doing is to get a to the power of 5 and a to the power of 6 plus b to the power of 6, right, and we can actually get a to the power of 6 plus b to the power of 6 if we should square a to the power of 3 plus b to the power of 3 yes So recall that a to the power of 3 plus b to the power of 3 is equal to 5 over 2. Yes, I wrote it. Somewhere, recall now. Let's continue from here. Let us continue. Let us continue. Okay, so we are Going to square, okay. Let's square this. So if we do that, we have a to the power of 3 plus b to the three squared is equal to five over two squared. Perfect. Now when you have two numbers squared You know how it is now, so it's going to be a to the power of 3 squared, which is going to be a cubed squared plus b cubed squared b cubed squared plus 2 what a to the power of 3 b to the power of 3 2 times a to the power of 3 b to the power of 3 is equal to now 5 squared, which gives us 25, and 2 squared, which gives us 4. Okay, so we a to the power of 3 times 2 gives you 6, okay, plus b to the power of 3 times 2 we give you 6, which is what we are looking for, okay, plus 2 a to the a cubed plus b cubed set as b raised to the power of 3 is equal to 25 over 4, so we have a to the power of 6 plus b to the power of 6 plus 2 bracket recall that our a b is minus 1 over 2, right, minus 1 over 2 raised to the power of 3 is equal to 25 over 4, please pardon the shadow, okay Pardon the shadow, alright, so we continue: a to the power of 6 plus b to the power 6 plus 2, bracket now minus 1 to the power of 3, we give you minus 1 and 2 to The power of 3 we give you 8 is equal to 25 over 4, so we have a to the power of 6. plus b to the power of 6 now 2 times minus and let's divide through we have 4, so it's going to be minus 1 over 4 minus 1 over 4, which is equal to 25 over 4. Perfect, so we have a to the power of 6 plus b to the power of 6 is equal to 25. over 4 now minus 1 over 4 we cross to become plus 1 over 4 all right now we have that a to the power of 6 plus b to the power of 6 is equal to we have the same denominator, so we can just add the exponents: 25 plus 1 gives us 26. Now if we reduce this: we are going to have what 13 over 2 in its lowest term now this is the main thing, but before that, if you've watched this far, that means that you truly enjoy what I'm doing, so please subscribe to the channel and on Your notification settings so that we notify you anytime I upload a new video Thank you so much. Now back to what we were doing, we have gotten our A to the six to the power of 6 and we have obtained a to the power of 5, so let's multiply them together to get what we are asked to get, so we are going to solve now, so this is a to the power of 5 plus b to the power of 5, so we are going to multiply equation 5 and equation 6; so if we do that, we'll have a to the power of 5 plus b to the power of 5 multiplied by a to the power of 6 plus b to the power of 6 is equal to 19 over 4 multiplied by 13 over 2, okay, so it's going to be 19 over 4 multiplied by 13 over 2 multiplied by 13 over 2, okay, all right, now let's Let's expand: if we do that, we have a to the power of 5 times a to the power of 6 we now give us a to the power of 5 plus 6 which is 11 plus a to the 5 times b to the power of 6 is a to the power of 5 b to the power of 6 b^5 a^6 b^5 a 6 plus b to the power of 5 times b to the power of 6, which is b to the power of 11 is equal to now 19 times 13. Let's solve that. 3 times 9 is 27. 3 times 1 is 3. plus 2 is 5, 1 times 9 is 9, and 1 times 1 is 1, so we are going to have 247, so it's going to be 247 over 8. All right, so we have a to the power of 11 plus b to the power of 11 plus now we can factorize a to the power of 5 b to the power of 5 here and when we open the bracket, when this is divided by this one, b will remain, and when this is divided By this one, a will remain equal to 247 over 8. All right, so we continue. So we now have a to the the power of 11 plus b to the power of 11 plus this same thing as was a b raised to the power of 5 right multiplied by b plus a is equal to 247 over 8, so we have a to the power of 11 plus b to the power of 11 plus now our a b is minus 1 over 2 minus 1 over 2 raised to the power of 5 and b plus a is 1 is equal to 247 over 8, all right, so we have a to the power of 11 plus b to the power of 11 now minus 1 to the power of 5 is still minus 1, so plus in brackets minus 1 over 2 to the the fifth power of 2 is 32, which is equal to 247 over 8, so a to the eleventh power plus b to the eleventh power should be equal to 247 over 8 now minus this we cross to become plus 1 over 32 now we solve that a The power of 11 plus b to the power of 11 is equal to now the LCM of this is what 32 now um when divided by 8 you have 4, 4 times 2 is 8, 4 times 7 is 28, 2 times 4 is 8, 7 times 4 is 28, 4 times 7 is 28, 4 times 4 is 16, plus 2 is 18 4 times 2 is 8, plus 1 is 9, so we have 9, 8, 8. Okay, plus 32 divided by 32, which gives us 1. 1 times 1 is 1, so we have a to the power of 11 plus b to the power of 11 equals 989 over 32. Thank you so much for watching. You have a lovely day and