Math OLYMPIAD| a Nice exponential problem

Welcome back, my lovely viewers. Let's solve this together. Okay, when you have something like this. The first thing we need to do is let's remove this square root; it actually doesn't make The question now appears ambiguous. We all know that the square root of x is the same thing as x to the power of one over two, right? Yes, so let's do that here. Let's remove the square root sign. that means we can rewrite this as 2 to the power of one half times x plus what two all right um now recall um using the law of indices when you have a to the power of m and your number is also raised to another power; the law of indices states that to obtain one power that we just have to multiply the two powers together, so we are going to write m times n okay so that simply means that we can rewrite this as two to the power of one half times what x plus two okay is equal to one over now this 32 can be written as two to the power of five two to the power of five, all right? So what do we do in indices? The best thing you can do for Yourselves is if you can actually put whatever you have, the numbers you have, to be on the same base. If once you're able to get them to share the same base, then your solution is very close. So this is one over two to the power of five; I want to bring this two to the power of five up. and the index indicates that one over x; if I want to eliminate this one over, I am going to introduce negative sign to the power of my denominator, so that means that it's going to be x to the power of minus one because the power of x is one, so I will just introduce a negative sign Okay, so that means I can write this as two to the power of one half times x plus two. is equal to two to the power of minus five, okay. All right, let's continue. Now at this point, notice that they are already on the same base, so that means the powers are equal once the bases are equal The powers are equal, which means we can now write that half bracket x plus two is equal to two. Sorry, it's equal to the power of this one, which is minus five. Okay, so let's open these brackets. Let's open the brackets: half times x is what x over two plus two times half is the same thing as what One right is equal to minus five. All right, now let's transfer this one to the other side of the equation, so let's subtract one from both sides of the equation, then we'll have that x over two is equal to minus five minus one, right? Yes, so we now have that x over two is equal to minus six. five minus one is minus six, right? Minus six. Okay, remember that what we are looking for is what the value of x at this point, you know that this is the same thing as minus six over one we haven't really changed anything, so when we cross-multiply, we'll have that two times minus Six we give us minus 12 is equal to x times one we give us x and this is the final solution thank Thank you so much for watching