Solve (frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}})*(x^2y+xy^2/x^2-y^2)

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

Evaluate

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Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{-2}-y^{-2}}{x^{-1}+y^{-1}}.

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

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

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

Express \frac{1}{y}x as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y}{y}.

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

Since \frac{y}{y} and \frac{x}{y} have the same denominator, add them by adding their numerators.

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

Calculate x to the power of 1 and get x.

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

Express \frac{y+x}{y}x as a single fraction.

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

Factor the expressions that are not already factored in \frac{x^{2}y+xy^{2}}{x^{2}-y^{2}}.

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

Cancel out x+y in both numerator and denominator.

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

Divide -y^{-2}x^{2}+1 by \frac{\left(y+x\right)x}{y} by multiplying -y^{-2}x^{2}+1 by the reciprocal of \frac{\left(y+x\right)x}{y}.

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

Multiply \frac{\left(-y^{-2}x^{2}+1\right)y}{\left(y+x\right)x} times \frac{xy}{x-y} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x in both numerator and denominator.

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

Multiply y and y to get y^{2}.

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

Use the distributive property to multiply -y^{-2}x^{2}+1 by y^{2}.

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

Multiply y^{-2} and y^{2} to get 1.

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

Consider \left(x+y\right)\left(x-y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Extract the negative sign in -x^{2}+y^{2}.

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

Cancel out x^{2}-y^{2} in both numerator and denominator.

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Factor

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