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

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

Step 1

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

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

Step 2

Simplify the denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Step 3

Factor $\frac{y + x}{x y}$ out of $\frac{1}{\frac{y + x}{x y} (x + y)}$ .

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

Step 4

Multiply $\frac{x y}{y + x} \cdot \frac{1}{x + y}$ .

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

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

Reorder terms.

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

Raise $x + y$ to the power of $1$ .

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

Raise $x + y$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

$\frac{x y}{{(x + y)}^{2}}$