# Exercise ID121

Algebra → Exponents → Powers
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Evaluate the following expression:

${a}^{-n}{\left({a}^{n}-1\right)}^{-1}-2{\left({a}^{2n}-1\right)}^{-1}+{a}^{-n}{\left({a}^{n}+1\right)}^{-1}$

${a}^{-n}{\left({a}^{n}-1\right)}^{{-}{1}}-2{\left({a}^{2n}-1\right)}^{{-}{1}}+{a}^{{-}{n}}{\left({a}^{n}+1\right)}^{{-}{1}}$ $=\frac{1}{{a}^{n}}·\frac{1}{{a}^{n}-1}-\frac{2}{{{a}}^{2n}{-}{1}}+\frac{1}{{a}^{n}}·\frac{1}{{a}^{n}+1}$ $=\frac{1}{{a}^{n}\left({a}^{n}-1\right)}-\frac{2}{\left({a}^{n}-1\right)\left({a}^{n}+1\right)}+\frac{1}{{a}^{n}\left({a}^{n}+1\right)}$ $=\frac{{a}^{n}+1-2{a}^{n}+\left({a}^{n}-1\right)}{{a}^{n}\left({a}^{n}-1\right)\left({a}^{n}+1\right)}$ $=\frac{2{a}^{n}-2{a}^{n}+1-1}{{a}^{n}\left({a}^{n}-1\right)\left({a}^{n}+1\right)}$ $=\frac{0}{{a}^{n}\left({a}^{n}-1\right)\left({a}^{n}+1\right)}$ $=0$
${a}^{-n}{\left({a}^{n}-1\right)}^{-1}-2{\left({a}^{2n}-1\right)}^{-1}+{a}^{-n}{\left({a}^{n}+1\right)}^{-1}=0$

${a}^{-n}=\frac{1}{{a}^{n}}$

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