Divide (x^-1+y^-1)/(x-y)

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

Step 1

Simplify the numerator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x}$ by $\frac{y}{y}$ .

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

Multiply $\frac{1}{y}$ by $\frac{x}{x}$ .

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{y + x}{x y}$ by $\frac{1}{x - y}$ .

$\frac{y + x}{x y (x - y)}$

Step 4

Split the fraction $\frac{y + x}{x y (x - y)}$ into two fractions.

$\frac{y}{x y (x - y)} + \frac{x}{x y (x - y)}$

Step 5

Cancel the common factor of $y$ .

Cancel the common factor.

$\frac{y}{x y (x - y)} + \frac{x}{x y (x - y)}$

Rewrite the expression.

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

Step 6

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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