A trending Nice Math Olympiad exponential question | how to solve for x

Hi everyone, this is JJ's online math class. Welcome to today's lesson. Today we are asked to find the value of x in this particular olympiad equation, okay, the first thing is let's simplify this particular question, this particular side—the left-hand side—and to simplify that, we can We can write 9 as 3 squared, and we can write 4 as 2 squared. All right, everything raised to the power. 9 over 4 is equal to x times root 6. Now, 3 squared over 2 squared can be written as When you have a squared over b squared, you can write this as a over b, all squared. a divided by b squared, okay? So let's do the same here. So we can write this as 3 over 2. x raised to the power of 9 over 4 is equal to the square root of 6, okay, all right Now we can express this as 9 over 4; 4 divides into 9, and we're going to have 2 with a remainder of 1, right? so it means that this is the same as 2 plus 1 over 4, so let's replace this with 2 plus 1 greater than 4, so we can write this as 3 over 2 squared raised to the power of 2 plus 1 over 4, okay Let's proceed. Recall the law of indices that states that a raised to the power of m raised to the power of n is equal to a raised to the power of m times n. Now let's relate this to what we have here, this particular 2 is to be used to multiply this and this, so we are going Multiply this and this by this 2, so it's going to be 3 over 2 raised to the power of 2 times 2. We have 4 plus 2 times 1 over 4, which is 2 over 4, okay, is equal to x root 6; sorry, this is also equal. to the sixth root of x, okay, all right, now we can write this as what, 3 over 2 raised to the power of 4 plus this divided by itself one, if we devise for we have to raise the power of 4 plus 1 over 2 okay is equal to x root 6 recall that a to the power of b plus c Yes, it is the same as a to the power of b times a to the power of c using the law of indices. That means we can rewrite the left-hand side of this equation as 3 over 2 to the power of 4. multiplied by 3 over 2 to the power of one half, okay, is equal to x times the square root of 6. All right, now recall also that x to the power of one half is the same thing as the square root of x, okay, so we can write this as 3 over 4, 3 over 3 divided by 2 to the power of 4 multiplied by the square root of 3 over 2 is equal to x times the square root of 6, okay? Now there's something I want to do—I want us to find the value of x. We need to remove this root 6 right away; the only way to remove this root 6 is if I can get root 6. this side of the equation and use it to cancel this, and the only way to introduce that here is If I multiply this by 2 and this by 2, okay, so I mean, I multiply this by 2 over 2, I've really Not multiplied by anything; that is, if I have 3 over 2 times 2 over 2, this will still cancel. You still have 3 over 2, so I haven't really introduced anything yet. Let's do that. and if we do that we are going to have 3 over 2 to the power of 4 multiplied by square root of 3 times 2 gives us 6, 2 times 2 gives us 4, which is equal to x root 6. All right, now I've gotten what I want to go ahead and separate this, because according to the rule of sod, the rule of sod states that The square root of p over q is equal to the square root of p over the square root of q, so we are going to apply that rule here. Now let's do it. so we have 3 over 2 raised to the power of 4, okay, times the square root of 6 over the square root of 4 is equal to x times the square root of 6; now, 3 to the power of 4 gives us 81, and 2 to the power of 4 gives us 16. √6 divided by √4 gives us 2, which is equal to x√6. Now let's multiply both multiply both sides of this equation by 2 over root 6, yes, let's multiply both sides by 2 over root 6 So if we do that, we have 81 over 16 times root 6 over 2 times 2 over root 6, okay, is equal to x. root 6 times 2 over root 6. All right, root 6 we cancel root 6, and 2 we cancel 2 as well, so We are left with what 81 over 16 equals now; here, the square root of 6 cancels the square root of 6. and we are left with 2x. All right, now let's multiply both sides of this equation by 1详细的 2 divided by half, so it's going to be 81 over 16 times half equals 2 times half, so 2 we cancel. then here 81 times 1 is 81, 16 times 2 is 32, is equal to x, therefore the value of x is equal to 81 over 32, sorry if this is the first time you're coming across this channel, please don't Don't forget to subscribe to the channel and then click on your notification bell so that you'll be notified. Whenever I upload a new video, please don't forget to leave your comments—your comments mean a lot to me because It helps me to know how you feel about what I'm doing, any corrections you think I need to make, and everything else. So in all, I say thank you for watching, and I wish to see you next time. Bye.