Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Reorder the factors of $y x$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

Cancel the common factor of $x + y$ and $y + x$ .

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Step 4.1

Reorder terms.

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Divide $x y$ by $1$ .

$x y$