# Exercise ID125

Algebra → Exponents → Powers
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Evaluate the following expression:

${\left(\frac{{x}^{-1}+{y}^{-1}}{y{x}^{-1}+x{y}^{-1}}\right)}^{-1}+{\left(\frac{{x}^{-1}+{y}^{-1}}{2}\right)}^{-1}-\frac{{x}^{-1}-{y}^{-1}}{{x}^{-1}·{y}^{-1}}$

${\left(\frac{{x}^{-1}+{y}^{-1}}{y{x}^{-1}+x{y}^{-1}}\right)}^{-1}+{\left(\frac{{x}^{-1}+{y}^{-1}}{2}\right)}^{-1}-\frac{{x}^{-1}-{y}^{-1}}{{x}^{-1}·{y}^{-1}}$ $=\frac{y{x}^{-1}+x{y}^{-1}}{{x}^{-1}+{y}^{-1}}+\frac{2}{{x}^{-1}+{y}^{-1}}-\frac{\frac{1}{x}-\frac{1}{y}}{\frac{1}{xy}}$ $=\frac{\frac{y}{x}+\frac{x}{y}}{\frac{1}{x}+\frac{1}{y}}+\frac{2}{\frac{1}{x}+\frac{1}{y}}-\frac{\frac{y-x}{xy}}{\frac{1}{xy}}$ $=\frac{\frac{{y}^{2}+{x}^{2}}{xy}}{\frac{y+x}{xy}}+\frac{2}{\frac{y+x}{xy}}-\frac{{x}{y}\left(y-x\right)}{{x}{y}}$ $=\frac{xy\left({x}^{2}+{y}^{2}\right)}{xy\left(x+y\right)}+\frac{2xy}{x+y}+x-y$ $=\frac{xy\left({x}^{2}+{y}^{2}\right)+2{\left(xy\right)}^{2}+xy\left(x+y\right)\left(x-y\right)}{xy\left(x+y\right)}$ $=\frac{{x}{y}\left[\left({x}^{2}+{y}^{2}\right){+}{2}{x}{y}+\left(x+y\right)\left(x-y\right)\right]}{{x}{y}\left(x+y\right)}$ $=\frac{{\left(x+y\right)}^{2}+\left(x+y\right)\left(x-y\right)}{x+y}$ $=\frac{\left(x+y\right)\left(x{+}{y}+x{-}{y}\right)}{\left(x+y\right)}$ $=2x$
${\left(\frac{{x}^{-1}+{y}^{-1}}{y{x}^{-1}+x{y}^{-1}}\right)}^{-1}+{\left(\frac{{x}^{-1}+{y}^{-1}}{2}\right)}^{-1}-\frac{{x}^{-1}-{y}^{-1}}{{x}^{-1}·{y}^{-1}}=2x$

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

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