Solve x^-1-y^-1/x^-2-y^-2

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored.

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

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

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

Expand the expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cancel out y in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out -x+y in both numerator and denominator.

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