# Exercise ID124

Algebra → Exponents → Powers
[Level: ] [Number of helps: 1] [Number of pictures: 0] [Number of steps: 8] [Number of characters: 0]

Evaluate the following expression:

$\frac{{a}^{n}-{a}^{-n}}{{a}^{n}+{a}^{-n}+2}$

$\frac{{a}^{n}-{a}^{-n}}{{a}^{n}+{a}^{-n}+2}$ $=\frac{{a}^{n}-\frac{1}{{a}^{n}}}{{a}^{n}+\frac{1}{{a}^{n}}+2}$ $=\frac{\frac{{a}^{n}·{a}^{n}-1}{{a}^{n}}}{\frac{{a}^{n}·{a}^{n}+1+2{a}^{n}}{{a}^{n}}}$ $=\frac{\frac{{a}^{2n}-1}{{a}^{n}}}{\frac{{a}^{2n}+2{a}^{n}+1}{{a}^{n}}}$ $=\frac{{{a}}^{n}\left({a}^{2n}-1\right)}{{{a}}^{n}\left({a}^{2n}+2{a}^{n}+1\right)}$ ${=}\frac{{a}^{2n}-1}{{a}^{2n}+2{a}^{n}+1}$ $=\frac{\left({a}^{n}-1\right)\left({a}^{n}+1\right)}{{\left({a}^{n}+1\right)}^{{2}}}$ $=\frac{{a}^{n}-1}{{a}^{n}+1}$
HELP AVAILABLE!
$\frac{{a}^{n}-{a}^{-n}}{{a}^{n}+{a}^{-n}+2}=\frac{{a}^{n}-1}{{a}^{n}+1}$

${a}^{-n}=\frac{1}{{a}^{n}};{\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}$

${a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right)$

${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$