Evaluating the value of exponential expression

Question

What is the value of:

$\frac{2a}{a^{x-y}-1}+\frac{2a}{a^{y-x}-1}$

I tried this:

$(\frac{a^{x-y}-1}{2a})^{-1}+(\frac{a^{y-x}-1}{2a})^{-1}$

$\frac{({a^{x-y}-1})^{-1}+({a^{y-x}-1})^{-1}}{2a^{-1}}$

But then I was stuck... any ideas?

Answer 1

Hint: the second fraction can be written $$\frac{2a}{a^{y-x}-1}=\frac{(2a)a^{x-y}}{1-a^{x-y}}$$ and now you can collect your two fractions over a common denominator.

Answer 2

we have $\left(\frac{1}{a^{x-y}-1}+\frac{1}{\frac{1}{a^{x-y}}-1}\right)$= $2a\left(\frac{1}{a^{x-y}-1}-\frac{a^{x-y}}{a^{x-y}-1}\right)=-2a$

Answer 3

HINT:

$$\frac1{a^{x-y}-1}=\frac1{\dfrac{a^x}{a^y}-1}=\frac{a^y}{a^x-a^y}$$