Solve x^-1+y^-1/x+y-x^-1-y^-1/x-y

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

Evaluate

$$\frac{2}{xy}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x^{-1}+y^{-1}}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x^{-1}-y^{-1}}{x-y} times \frac{x+y}{x+y}.

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

Since \frac{\left(x^{-1}+y^{-1}\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{\left(x^{-1}-y^{-1}\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{-1}+y^{-1}\right)\left(x-y\right)-\left(x^{-1}-y^{-1}\right)\left(x+y\right).

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

Combine like terms in 1-\frac{1}{x}y+\frac{1}{y}x-1-1-\frac{1}{x}y+\frac{1}{y}x+1.

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

Expand \left(x+y\right)\left(x-y\right).

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

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

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

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

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

Express -2\times \frac{1}{x} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{2x}{y} times \frac{x}{x}. Multiply \frac{-2y}{x} times \frac{y}{y}.

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

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

$$\frac{\frac{2xx-2yy}{xy}}{x^{2}-y^{2}}$$

Do the multiplications in 2xx-2yy.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$\frac{2}{xy}$$

Expand

$$\frac{2}{xy}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x^{-1}+y^{-1}}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x^{-1}-y^{-1}}{x-y} times \frac{x+y}{x+y}.

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

Since \frac{\left(x^{-1}+y^{-1}\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)} and \frac{\left(x^{-1}-y^{-1}\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{-1}+y^{-1}\right)\left(x-y\right)-\left(x^{-1}-y^{-1}\right)\left(x+y\right).

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

Combine like terms in 1-\frac{1}{x}y+\frac{1}{y}x-1-1-\frac{1}{x}y+\frac{1}{y}x+1.

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

Expand \left(x+y\right)\left(x-y\right).

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

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

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

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

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

Express -2\times \frac{1}{x} as a single fraction.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and x is xy. Multiply \frac{2x}{y} times \frac{x}{x}. Multiply \frac{-2y}{x} times \frac{y}{y}.

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

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

$$\frac{\frac{2xx-2yy}{xy}}{x^{2}-y^{2}}$$

Do the multiplications in 2xx-2yy.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

$$\frac{2}{xy}$$