Harvard University interview algebra math question| algebra simplification.

Watch this video to the end, because I'll be giving a step-by-step method on how to solve it. This kind of problem—okay, now first let us simplify root 12. Now let me tell you the easiest way to do this no matter how large that number is; if you want to simplify a sword, write that sword What is the smallest number that can divide both of these? We go with that two now. When 12 is we have six when six is divided by two again because that is the smallest number that can divide six even though three can divide it but we go with two okay now when six is divided by Two we have three now. When three divides itself, we have one now. Remember that this is square root. So we're going to take this in pairs, okay? Since this is the only pair we can see here. two times two gives us four, so it means that four times three is the easiest way to express it. 12 so that we can simplify it further, okay, so I'm not going to write this as the square root of four multiplied by the square root of three, because when you have the square root of a times b, it is equal to the square the square root of a multiplied by the square root of big, so 12 can be written as the square root of four multiplied by the square root of three, you're going to see why soon, divided by now root 15 can't Does it have any perfect squares in it? So we can just express it as the square root of five times three. there is no other way we can express that, so the square root of five multiplied by the square root of three plus the square root of three, okay, all these things raised to the power of so, okay, now we can write this to be equal to the square root of four is two, can you see why we decided to express this in terms of because we can get the square root of four; four is a perfect square; now, the square root of four is two. So we have two over root three divided by now they have three in common, right? So we can bring three. out and we have root five and what that is root five multiplied by root three divided by root three gives us root five, right? Then plus root three divided by root three will give us one. Now this is what we have, everything raised to the power of 12. Okay, let me come here now. You see that we have root three here and root three here, so we can cancel them. So what do we have now? has two over the square root of five plus one raised to the power of 12, okay? Now remember that This is a sword, and according to the law of swords, it is improper for you to leave the sword as your denominator. So since root five is the sword, we can, uh, we are going to rationalize this fraction. by multiplying the numerator and denominator by the conjugate of this denominator now the The conjugate of this denominator is root five minus one, so this time we change to minus if this is The square root of five plus or minus one; then the conjugate will be the square root of five plus one, so in this case it is plus. So this is what I mean: we are going to multiply the numerator by root five minus one, the conjugate. of this, and also multiply the denominator by the same thing, okay, root five minus one That is how to rationalize. Okay, everything raised to the power of 12. Now we continue to solve, so. This is what we have: two brackets, root five minus one, divide that by. Now when you have a plus b multiplied by a minus b; this is the difference of two squares, right? So this is the same thing as a. squared minus b squared, so our a here is root five, so this expression root five squared minus our b is one, one squared; remember that the whole of this is raised to the power of 12 now we are Even as we keep simplifying, we haven't even begun solving yet, but trust me, once we start solving It's more interesting, okay. All right, so this place we have two root five minus one. divided by, then this, we can see, this we have five minus one squared is still one, everything raised to the power of 12, okay, so this is equal to two brackets, root five minus one, divided by four because five minus one is four, everything raised to the power of so permits me to continue From this side of the board, if this is the first time you've seen this amazing channel Click the subscription button and turn on your notification bell so I don't miss our amazing videos, we upload our videos every single day, like two or three times daily, so tune in at your convenience. Convenience to watch our new uploads and give us a thumbs up if you're actually enjoying this video Thank you very much. Now at this point, this is what we have. So, two can divide this to give us two so in summary we have root five minus one divided by two raised to the power of 12 Now, from this path, whenever you see something in this form, okay, to solve this, call whatever. you have inside the bracket anything, so I'll call mine x; you can start to call yours y or anything of your choice so we can say let x be equal to the square root of five minus one divided by two Now when you do that, that means you can say this is equal to x to the power of 12, right? Yes, now I'm going to try to find the value of x squared. I'll explain later why. I'm trying to get that value, okay? All right, now let's go. So we are going to cross-multiply. If we cross-multiply, we have x times two, which gives us 2x. Okay, that means this narrative All right, now when this is multiplied by one, we have root five minus one, which is equal to root five minus one. One, okay, it means we need to clean everything from here. All right, now you'll transfer your constants to this side, as I said, it's a step-by-step approach, so transfer this minus one to the other side, your constant, so we have two x minus one; we cross to become plus one. equal to root five now, because we want to eliminate this result, we square this side, right? Whatever we do to this side, we do the same to the other side, so let us square this side. also two x plus one squared, okay, is equal to root five squared, all right. Now remember that When you have a plus b squared, that is equal to a squared plus two a b plus b squared, all right. That means we have two x squared plus two multiplied by two x multiplied by one. plus one squared is equal to now this you can see this you have only five here now two squared is four x squared plus two times two x times one gives us four x plus one squared is one is equal to five, so we have four x squared plus four x plus one minus five, we cross to become plus five is equal to zero, as I said. My intention here is to obtain the value of x squared, and I'm going to explain why soon okay so we have four x squared plus four x plus oh minus one because one minus Five is minus four, and plus times minus is minus, so we have minus four is equal to zero. Now notice that we have four four four, we can decide to divide through by four to simplify this further so if we divide through by four, we have this divided by four, divided by four, divided by four divided by four, so this will cancel this; we have x squared, right? So we have x squared plus this. We divide it by itself; we have only x minus this. We divide this, and we have one is equal to zero now at. At this point, remember, I told you my intention is to get the şart of x squared, so I'm going to transfer everything to the other side, leaving x squared alone, so if I do that, I will x squared equals minus x; plus x crosses over to become minus x minus one, which crosses over become plus one, so it means that x squared is equal to plus one minus x. Now this is a very Important equation: let's call this Equation One, and I'm going to write it down somewhere in case I Clean this off so I won't forget: we have x squared equals one minus x. Now let me explain why. I am trying to get the value of x squared. The first thing you need to know is that we are looking for x to the power of 12; remember, we said let this be equal to x, so that means we are looking for x to the power of 12; now, if we use the law of indices or exponents, note or remember that a to The power of m raised to the power of n is equal to a to the power of m times n, okay, yeah, so. simply means that 12 can be expressed in terms of its product of factors as the product of its Now let's look at the factors. We need to know that x cubed raised to the power of 2 We give us 6 right now; if I should square this as well, I'll get x to the power of 12. Yes, 3 times 3 is 9, 9 times 2 is 18, so it means that if I can get x to the power of 3, I'll square it. it, and then I'll square it again to get to what we're looking for. Do you understand now? the essence of x squared remember that if I multiply x squared by x to the power of 1 I will get x to the power of 3, yes, right, so that's it; once I do this to this, I'll get my x to the power of 3 and I'll use it as a base to get to my destination. Let's do it together, so that it It won't be difficult; so, as I said, we are going to multiply this side by x to the power of 1. And you know, in mathematics, whatever you do to one side, you do to the other side. To balance it out, okay, so we are going to have x squared multiplied by x is equal to 1 minus x. multiplied by x as well, so here we have x to the power; remember that x is the same as x to the power Of 1, so we pick one base and we add the powers 2 plus 1, which is 3, equal to x times 1, which is x. minus x times x is x squared; now remember I said that x squared is the most important equation. So far, this is why: because we are simplifying without a calculator, anywhere we see x squared Excuse me, we are going to replace it with 1 minus x, okay? So that means x to the power of 3. is equal to x minus x squared, which is 1 minus x, okay? So we have x to the power of 3 equal to x. minus times 1 is minus 1; minus times minus x is plus x. Okay, so we have x to the power of 3 is equal to x plus x is 2x minus 1, now we have gotten x to the power of 3, remember our intention we are going to square this to get x to the power of 6, okay, and square it again to get to our not just reaching the destination, but so many things are involved Why do we get to the destination? Okay, so we have x to the power of 3. Now we are going to square it. both sides to get x to the power of 6, so we have x to the power of 3 squared is equal to 2x minus 1 squared, okay, so x to the power of 6 is equal to. Remember that a minus b squared is equal to a minus 2ab plus b squared, so this will be 2x squared minus 2ab, 2 multiplied by 2x multiplied by 1 okay plus b squared plus 1 squared, so x to the power of 6 is 2 squared is 4 x squared is x squared minus 2 times 2x is 4x plus 1, okay. Recall anywhere we see x squared. We replace it with 1 minus x, so we have x to the power of 6 equals 4 times (1 minus x). minus 4x plus 1, okay. All right, now let's continue. So we have that x to the power of 6 is equal to 4 times 1 is 4 minus 4 times x is 4x, okay. Now minus 4x plus 1, okay. So x to the power of 6 is equal to 4 plus 1 gives us 5 minus 4x minus 4x is minus 8x. Now we are going to square x to the 6 in order to get x to the power of 12, remember, all right, so we have x to the power of 6 squared. So we also square this one as well. Okay, so we have x to the power of 12 is equal to 5 squared. minus 2 multiplied by 5 multiplied by 8x plus 8x squared, so we have our x to the power of 12 equals 25 minus now 2 times 5 is 10, 10 times 8 is 80, so we have minus 80x plus x squared equals 64x squared, all right? Now remember that we replace our x squared with 1 minus x, okay, so we have 25 minus 80x plus 64 brackets 1 minus x, okay, so x to the power of 12 is equal to 25 minus 80x plus 64 times 1 is 64, 64 times minus x is minus 64x, now we continue From here, okay, we are almost at the end of this journey, so we have x to the power of 12 is equal To combine like terms, 25 plus 64 gives us 89, so we have 89 now minus 80 minus 64 is minus 114 minus 114x, okay. Now it's time to substitute our x, but remember that we got our x as root 5. minus 1 over 2, so we are going to replace that here, so we have root 5 minus 1 over 2 raised to the power of 12 is equal to 89 minus 114 times the quantity: the square root of 5 minus 1 over 2. Yeah, we are I'm going to use the simplified version, not this one. Okay, so we have root 5 minus 1 over 2, all right. Now we have, um, this to be what I don't want to keep repeating, 89 minus now, so we divide. This is 144, sorry, 144 now, so we divide this to give us 72, okay? I'm sorry, so we divide this. to give us 72 we have 72 times the square root of 5 minus 1, all right, so this is equal to 89 minus 72 times root 5 we get 72 root 5, all right, minus 1 minus 72 times minus 1 we get plus 72, so we Solve this: 9 plus 1 gives us 11, we write down 1, carry 1. Okay, now 1 plus 7 is 8, 8 plus 8 is 16. Okay, now therefore, therefore, root 12 divided by root 15 plus root 3, raised to the power of 12. can be expressed as 161 minus 72 root 5 in its simplest form without the use of a calculator Thank you so much for watching, see you in my next video, bye!