Question 231057
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.  {{{(x^a)^b = x^(a*b)}}}


Concept 2.  {{{x^(a-b) = x^a/x^b}}}


Concept 3.  {{{(a/b) / (c/d) = (a/b) * (d/c) = (a*d) / (b*c)}}} 


Back to your problem which is to simplify:


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


Use concept 1 to make your equation equal to:


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


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:


{{{(x^(a^2-b^2)) / (x^(a^2-4b^2))}}}


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


{{{(x^(a^2)/x^(b^2)) / (x^(a^2)/x^(4b^2))}}}


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


{{{(x^(a^2) * x^(4b^2)) / (x^(b^2) * x^(a^2))}}}


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^(3b^2)}}} 


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^(3b^2)}}}