Simplify (y^-1+x^-1)/((xy)^-1)

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

Step 1

Move ${(x y)}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify each term.

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

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

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

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

Reorder the factors of $y x$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Cancel the common factor of $x y$ .

Cancel the common factor.

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

Rewrite the expression.

$x + y$