Question

[{x^(a/a-b)/x^(a/a+b)}/{x^(b/b-a)/x^(b/b+1)}]^(a+b)

Answer 1

$[( \frac{ x^{a/a-b} }{ x^{a/a+b} } )/( \frac{ x^{b/b-a} }{ x^{b/b+a} } )] ^{a+b} \\ \\ =[(x^{ \frac{a}{a-b} - \frac{a}{a+b} })/( x^{ \frac{b}{b-a} - \frac{b}{b+a} } )] ^{a+b} \\ \\ =[(x^{ \frac{a^{2}+ab-a^{2}+ab}{a^{2}-b^{2}} })/( x^{ \frac{b^{2}+ab-b^{2}+ab }{(b-a)(b+a)} } )] ^{a+b} \\ \\ =[(x^{ \frac{2ab}{a^{2}-b^{2}} })/( x^{ \frac{2ab }{(b^{2}-a^{2}} } )] ^{a+b} \\ \\ =(x^{ \frac{(2ab)(a+b)}{a^{2}-b^{2}} })/( x^{ \frac{(2ab)(a+b) }{(b^{2}-a^{2}} } )$

$=(x^{ \frac{2ab}{a-b} })/( x^{ \frac{2ab}{b-a} } ) \\ \\ =x^{2ab( \frac{1}{a-b} - \frac{1}{b-a} )} \\ \\ =x^{2ab( \frac{1}{a-b} + \frac{1}{a-b} )} \\ \\ =x^{ \frac{4ab}{a-b}}$

Note: (b/b+a) has been used instead of (b/b+1) as it is easy to do. if (b/b+1) is used, the expression would be a bit long, but it can be done.