A tricky exponent's problem that tests your mathematical thinking

Let's find the values of x and y. First of all, recall that a to the power m raised to the power of n can Also written as a to the power of m times n, right? Welcome back to my channel. So let's get on with this. I forgot to welcome you. So now, we can write this as 3 to the power of x squared, right? -2 to the power of y raised The square of 2 is equal to 17. Now, if you look closely, You are going to notice the difference of two squares. Now, for the difference of two squares, we have that a squared minus b squared is equal to a plus b multiplied by a minus b. Now in this case a is three to the power of x and our b is 2 to the power of y if you're enjoying this and This is the first time you're going to come to Please hit the subscription button for this channel and Please comment and let me know what you think about it. The video, all right, thank you very much. Now that means that we can rewrite this as three 2^x + 2^y multiplied by 3 to the power of x minus 2 y raised to the power of 17. Now, note that 17 is equal to 17 times 1, right? So, we can actually rewrite this as 3. x to the power of 2 plus y to the power of 2 multiplied by one y to the power of y is equal to 17 multiplied by 1. Now, this simply means that because this is definitely larger than this, because here you are adding 2 to the power of y to a number, but you are actually subtracting the same amount from the same number. Therefore, this is larger. So, 3 to the power of x plus 2 to the power of y is greater, right? And if it is greater than 3 to the power of x minus 2 to the power of y. we can now write that 3 to the power of x plus 2 to the power of y should be equal to the larger number 17. Now let's call this equation 1 3 to the power of x minus 2 x to the power of y is equal to 1 and let's call this equation 2. Now If you solve for that, let us add equation 1. to equation 2 So 3 to the power of x plus 3 to the power of x is 2 times 3 to the power of x. Now this plus minus this is 0. is equal to 17 plus 1, which is 18. Now let's divide both sides by 2. Divide both sides by 2. So we have 2 brackets 3 to the power x divided by 2 is equal to 18 divided by 2. Now this and this, we go so we have 3 to the power of x is equal to 2 we go into 18 9 times this is a very interesting math video, please drop Your comment if you have a faster way Please don't hesitate to share your solution in the comments Okay, so moving forward we have 3. x raised to the power of x equals 9 now let's write nine in base three, right? if we so that we can say three to the power of x is equal To date, nine is three times three. which is the same thing as three squared, so we can write this as three squared; now note that a to the power of m equals a to the power of n when a to the power of m is equal to a to the power of n This implies that m is equal to n. therefore we can say from here that this implies that x is equal to 2, so we have a value of x Now if you look at the oh I've deleted that box we can see from here that 3 to the power of x -2 to the power of y is equal to 1 and 3 to the power of x plus 2 to the power of y is 17 So, BOTS! 3 to the power of x plus 2 to the power of y is 17 Now x is 2, so we can say 3 to the power of 2 plus 2. to the power of y is 17, so now we have 9 plus 2 to the power of y is 17. We are Almost there. Okay, let's subtract. 9 from both sides, so we are going to have that 2 to the power of y is equal To 8 we can actually put 8 in base 2, okay but eight is two times two times two which is the same thing as two to the power of three So we can now write that two to the power of y is two. to the power of three, which implies that y is equal to three, so now x is equal to two and y equals three, the equation says x plus y So, we write that as 2 plus 3, which equals 5. And that is our answer. Thank you so much for watching, and I'll see you in my next video. Bye.