SOLUTION: Hi, I'm having problems simplifying this problem: (x^{a+b})^{a-b} ----------- ----- (x^{a-2b})^{a+2b} Some of the things I have tried: x^{2a-2b} ----------

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Hi, I'm having problems simplifying this problem: (x^{a+b})^{a-b} ----------- ----- (x^{a-2b})^{a+2b} Some of the things I have tried: x^{2a-2b} ----------       Log On


   

Question 231057: Hi, I'm having problems simplifying this problem:
(x^{a+b})^{a-b}
-----------------
(x^{a-2b})^{a+2b}
Some of the things I have tried:
x^{2a-2b}
---------- ---> x^2b
x^{2a-4b}
I'm also beginning to think it isn't possible to simplify any further than the original question. Help?
Thanks in advance.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
I'll take a crack at it.

Your problem is:

(x^{a+b})^{a-b}
-----------------
(x^{a-2b})^{a+2b}

A couple of basic concepts should help you. There is no relationship between any of the variables shown below and the ones in your problem.

Concept 1. %28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29

Concept 2. x%5E%28a-b%29+=+x%5Ea%2Fx%5Eb

Concept 3. %28a%2Fb%29+%2F+%28c%2Fd%29+=+%28a%2Fb%29+%2A+%28d%2Fc%29+=+%28a%2Ad%29+%2F+%28b%2Ac%29

Back to your problem which is to simplify:

%28%28x%5E%28a%2Bb%29%29%5E%28a-b%29%29+%2F+%28%28x%5E%28a-2b%29%29%5E%28a%2B2b%29%29

Use concept 1 to make your equation equal to:

%28x%5E%28%28a%2Bb%29%2A%28a-b%29%29%29+%2F+%28x%5E%28%28a-2b%29%2A%28a%2B2b%29%29%29

When you multiply (a+b) * (a-b) you get a^2 - b^2

When you multiply (a+2b) * (a-2b) you get a^2 - 4b^2

Your equation becomes:

%28x%5E%28a%5E2-b%5E2%29%29+%2F+%28x%5E%28a%5E2-4b%5E2%29%29

Using concept 2, we change this equation to be equivalent to:

%28x%5E%28a%5E2%29%2Fx%5E%28b%5E2%29%29+%2F+%28x%5E%28a%5E2%29%2Fx%5E%284b%5E2%29%29

Using concept 3, we change this equation to be equivalent to:

%28x%5E%28a%5E2%29+%2A+x%5E%284b%5E2%29%29+%2F+%28x%5E%28b%5E2%29+%2A+x%5E%28a%5E2%29%29

The x^(a^2) cancels out and the x^(4b^2) / x^(b^2) becomes:

x^(4b^2 - b^2) which becomes x^(3b^2) using concept 2.

You are left with:

x%5E%283b%5E2%29

as your answer.

To see if this is correct, we assign some values at random to the original equation and the final equation to see if they are equivalent.

I assigned values of:

a = 5, b = 2
and:
a = 2, b = 2
and:
a = 2, b = 3

and confirmed that the original equation and the final equation are equivalent.

I think the answer is good.

The answer is:

x%5E%283b%5E2%29