Algebra Examples

$\frac{2 x^{- 1} + 2 y^{- 1}}{9 {(x + y)}^{- 1}}$

Step 1

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

$\frac{(2 x^{- 1} + 2 y^{- 1}) (x + y)}{9}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{- 1} + 2 y^{- 1}$ .

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Step 2.1.1

Factor $2$ out of $2 x^{- 1}$ .

$\frac{(2 (x^{- 1}) + 2 y^{- 1}) (x + y)}{9}$

Step 2.1.2

Factor $2$ out of $2 y^{- 1}$ .

$\frac{(2 (x^{- 1}) + 2 (y^{- 1})) (x + y)}{9}$

Step 2.1.3

Factor $2$ out of $2 (x^{- 1}) + 2 (y^{- 1})$ .

$\frac{2 (x^{- 1} + y^{- 1}) (x + y)}{9}$

Step 2.2

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

$\frac{2 (\frac{1}{x} + y^{- 1}) (x + y)}{9}$

Step 2.3

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

$\frac{2 (\frac{1}{x} + \frac{1}{y}) (x + y)}{9}$

Step 2.4

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

$\frac{2 (\frac{1}{x} \cdot \frac{y}{y} + \frac{1}{y}) (x + y)}{9}$

Step 2.5

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

$\frac{2 (\frac{1}{x} \cdot \frac{y}{y} + \frac{1}{y} \cdot \frac{x}{x}) (x + y)}{9}$

Step 2.6

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

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Step 2.6.1

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

$\frac{2 (\frac{y}{x y} + \frac{1}{y} \cdot \frac{x}{x}) (x + y)}{9}$

Step 2.6.2

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

$\frac{2 (\frac{y}{x y} + \frac{x}{y x}) (x + y)}{9}$

Step 2.6.3

Reorder the factors of $y x$ .

$\frac{2 (\frac{y}{x y} + \frac{x}{x y}) (x + y)}{9}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{2 \frac{y + x}{x y} (x + y)}{9}$

Step 2.8

Combine $2$ and $\frac{y + x}{x y}$ .

$\frac{\frac{2 (y + x)}{x y} (x + y)}{9}$

Step 3

Factor $\frac{2 (y + x)}{x y}$ out of $\frac{\frac{2 (y + x)}{x y} (x + y)}{9}$ .

$\frac{2 (y + x)}{x y} \cdot \frac{x + y}{9}$

Step 4

Multiply $\frac{2 (y + x)}{x y} \cdot \frac{x + y}{9}$ .

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Step 4.1

Multiply $\frac{2 (y + x)}{x y}$ by $\frac{x + y}{9}$ .

$\frac{2 (y + x) (x + y)}{x y \cdot 9}$

Step 4.2

Reorder terms.

$\frac{2 (x + y) (x + y)}{x y \cdot 9}$

Step 4.3

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

$\frac{2 ({(x + y)}^{1} (x + y))}{x y \cdot 9}$

Step 4.4

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

$\frac{2 ({(x + y)}^{1} {(x + y)}^{1})}{x y \cdot 9}$

Step 4.5

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

$\frac{2 {(x + y)}^{1 + 1}}{x y \cdot 9}$

Step 4.6

Add $1$ and $1$ .

$\frac{2 {(x + y)}^{2}}{x y \cdot 9}$

Step 5

Move $9$ to the left of $x y$ .

$\frac{2 {(x + y)}^{2}}{9 x y}$