Pre-Algebra Examples

$\frac{\frac{1}{2 x^{2}} + \frac{x}{4}}{\frac{1}{x + 2} - \frac{x}{2}}$

Step 1

Multiply the numerator and denominator of the fraction by $4 (x + 2) x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{1}{2 x^{2}} + \frac{x}{4}}{\frac{1}{x + 2} - \frac{x}{2}}$ by $\frac{4 (x + 2) x^{2}}{4 (x + 2) x^{2}}$ .

$\frac{4 (x + 2) x^{2}}{4 (x + 2) x^{2}} \cdot \frac{\frac{1}{2 x^{2}} + \frac{x}{4}}{\frac{1}{x + 2} - \frac{x}{2}}$

Step 1.2

Combine.

$\frac{4 (x + 2) x^{2} (\frac{1}{2 x^{2}} + \frac{x}{4})}{4 (x + 2) x^{2} (\frac{1}{x + 2} - \frac{x}{2})}$

Step 2

Apply the distributive property.

$\frac{4 (x + 2) x^{2} \frac{1}{2 x^{2}} + 4 (x + 2) x^{2} \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $2 x^{2}$ .

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Step 3.1.1

Factor $2 x^{2}$ out of $4 (x + 2) x^{2}$ .

$\frac{2 x^{2} (2 (x + 2)) \frac{1}{2 x^{2}} + 4 (x + 2) x^{2} \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.1.2

Cancel the common factor.

$\frac{2 x^{2} (2 (x + 2)) \frac{1}{2 x^{2}} + 4 (x + 2) x^{2} \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.1.3

Rewrite the expression.

$\frac{2 (x + 2) + 4 (x + 2) x^{2} \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Factor $4$ out of $4 (x + 2) x^{2}$ .

$\frac{2 (x + 2) + 4 ((x + 2) x^{2}) \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.2.2

Cancel the common factor.

$\frac{2 (x + 2) + 4 ((x + 2) x^{2}) \frac{x}{4}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.2.3

Rewrite the expression.

$\frac{2 (x + 2) + (x + 2) x^{2} x}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.3

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

$\frac{2 (x + 2) + (x + 2) (x^{1} x^{2})}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.4

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

$\frac{2 (x + 2) + (x + 2) x^{1 + 2}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.5

Add $1$ and $2$ .

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 (x + 2) x^{2} \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.6

Cancel the common factor of $x + 2$ .

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Step 3.6.1

Factor $x + 2$ out of $4 (x + 2) x^{2}$ .

$\frac{2 (x + 2) + (x + 2) x^{3}}{(x + 2) (4 x^{2}) \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.6.2

Cancel the common factor.

$\frac{2 (x + 2) + (x + 2) x^{3}}{(x + 2) (4 x^{2}) \frac{1}{x + 2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.6.3

Rewrite the expression.

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} + 4 (x + 2) x^{2} (- \frac{x}{2})}$

Step 3.7

Cancel the common factor of $2$ .

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Step 3.7.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} + 4 (x + 2) x^{2} \frac{- x}{2}}$

Step 3.7.2

Factor $2$ out of $4 (x + 2) x^{2}$ .

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} + 2 (2 (x + 2) x^{2}) \frac{- x}{2}}$

Step 3.7.3

Cancel the common factor.

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} + 2 (2 (x + 2) x^{2}) \frac{- x}{2}}$

Step 3.7.4

Rewrite the expression.

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} + 2 (x + 2) x^{2} (- x)}$

Step 3.8

Multiply $- 1$ by $2$ .

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} - 2 (x + 2) x^{2} x}$

Step 3.9

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

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} - 2 (x + 2) (x^{1} x^{2})}$

Step 3.10

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

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} - 2 (x + 2) x^{1 + 2}}$

Step 3.11

Add $1$ and $2$ .

$\frac{2 (x + 2) + (x + 2) x^{3}}{4 x^{2} - 2 (x + 2) x^{3}}$

Step 4

Factor $x + 2$ out of $2 (x + 2) + (x + 2) x^{3}$ .

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Step 4.1

Factor $x + 2$ out of $2 (x + 2)$ .

$\frac{(x + 2) \cdot 2 + (x + 2) x^{3}}{4 x^{2} - 2 (x + 2) x^{3}}$

Step 4.2

Factor $x + 2$ out of $(x + 2) x^{3}$ .

$\frac{(x + 2) \cdot 2 + (x + 2) (x^{3})}{4 x^{2} - 2 (x + 2) x^{3}}$

Step 4.3

Factor $x + 2$ out of $(x + 2) \cdot 2 + (x + 2) (x^{3})$ .

$\frac{(x + 2) (2 + x^{3})}{4 x^{2} - 2 (x + 2) x^{3}}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $2 x^{2}$ out of $4 x^{2} - 2 (x + 2) x^{3}$ .

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Step 5.1.1

Factor $2 x^{2}$ out of $4 x^{2}$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2) - 2 (x + 2) x^{3}}$

Step 5.1.2

Factor $2 x^{2}$ out of $- 2 (x + 2) x^{3}$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2) + 2 x^{2} (- (x + 2) x)}$

Step 5.1.3

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

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 - (x + 2) x)}$

Step 5.2

Apply the distributive property.

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 + (- x - 1 \cdot 2) x)}$

Step 5.3

Multiply $- 1$ by $2$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 + (- x - 2) x)}$

Step 5.4

Apply the distributive property.

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 - x \cdot x - 2 x)}$

Step 5.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.5.1

Move $x$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 - (x \cdot x) - 2 x)}$

Step 5.5.2

Multiply $x$ by $x$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (2 - x^{2} - 2 x)}$

Step 5.6

Reorder terms.

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- x^{2} - 2 x + 2)}$

Step 6

Simplify with factoring out.

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Step 6.1

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

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- (x^{2}) - 2 x + 2)}$

Step 6.2

Factor $- 1$ out of $- 2 x$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- (x^{2}) - (2 x) + 2)}$

Step 6.3

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

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- (x^{2} + 2 x) + 2)}$

Step 6.4

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- (x^{2} + 2 x) - 1 (- 2))}$

Step 6.5

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

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- (x^{2} + 2 x - 2))}$

Step 6.6

Rewrite negatives.

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Step 6.6.1

Rewrite $- (x^{2} + 2 x - 2)$ as $- 1 (x^{2} + 2 x - 2)$ .

$\frac{(x + 2) (2 + x^{3})}{2 x^{2} (- 1 (x^{2} + 2 x - 2))}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{(x + 2) (2 + x^{3})}{(2 x^{2}) (x^{2} + 2 x - 2)}$

Step 6.6.3

Reorder factors in $- \frac{(x + 2) (2 + x^{3})}{(2 x^{2}) (x^{2} + 2 x - 2)}$ .

$- \frac{(x + 2) (2 + x^{3})}{2 x^{2} (x^{2} + 2 x - 2)}$