Solve (x^-2-y^-2):(x^-1+y^-1)

$$( x ^ { - 2 } - y ^ { - 2 } ) : ( x ^ { - 1 } + y ^ { - 1 } )$$

Evaluate

$$\frac{y-x}{xy}$$

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

To raise \frac{x}{y} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

Since \frac{y^{2}}{y^{2}} and \frac{x^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Express \frac{1}{y}x^{2} as a single fraction.

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

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

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

Since \frac{xy}{y} and \frac{x^{2}}{y} have the same denominator, add them by adding their numerators.

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

Divide \frac{y^{2}-x^{2}}{y^{2}} by \frac{xy+x^{2}}{y} by multiplying \frac{y^{2}-x^{2}}{y^{2}} by the reciprocal of \frac{xy+x^{2}}{y}.

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

$$\frac{-x+y}{xy}$$

Expand

$$\frac{y-x}{xy}$$

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

To raise \frac{x}{y} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{y^{2}}{y^{2}}.

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

Since \frac{y^{2}}{y^{2}} and \frac{x^{2}}{y^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Express \frac{1}{y}x^{2} as a single fraction.

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

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

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

Since \frac{xy}{y} and \frac{x^{2}}{y} have the same denominator, add them by adding their numerators.

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

Divide \frac{y^{2}-x^{2}}{y^{2}} by \frac{xy+x^{2}}{y} by multiplying \frac{y^{2}-x^{2}}{y^{2}} by the reciprocal of \frac{xy+x^{2}}{y}.

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+y in both numerator and denominator.

$$\frac{-x+y}{xy}$$