Precalculus Examples

$\frac{(x - 4)}{(\frac{x}{4} - \frac{4}{x})}$

Step 1

Multiply the numerator and denominator of the fraction by $4 x$ .

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Step 1.1

Multiply $\frac{x - 4}{\frac{x}{4} - \frac{4}{x}}$ by $\frac{4 x}{4 x}$ .

$\frac{4 x}{4 x} \cdot \frac{x - 4}{\frac{x}{4} - \frac{4}{x}}$

Step 1.2

Combine.

$\frac{4 x (x - 4)}{4 x (\frac{x}{4} - \frac{4}{x})}$

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $4$ .

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Step 3.1.1

Factor $4$ out of $4 x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $x$ .

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Step 3.6.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

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

Step 3.6.2

Factor $x$ out of $4 x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Multiply $4$ by $- 4$ .

$\frac{4 x \cdot x + 4 x \cdot - 4}{x^{2} - 16}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Move $x$ .

$\frac{4 (x \cdot x) + 4 x \cdot - 4}{x^{2} - 16}$

Step 4.1.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 4 x \cdot - 4}{x^{2} - 16}$

Step 4.2

Multiply $- 4$ by $4$ .

$\frac{4 x^{2} - 16 x}{x^{2} - 16}$

Step 4.3

Factor $4 x$ out of $4 x^{2} - 16 x$ .

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Step 4.3.1

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

$\frac{4 x (x) - 16 x}{x^{2} - 16}$

Step 4.3.2

Factor $4 x$ out of $- 16 x$ .

$\frac{4 x (x) + 4 x (- 4)}{x^{2} - 16}$

Step 4.3.3

Factor $4 x$ out of $4 x (x) + 4 x (- 4)$ .

$\frac{4 x (x - 4)}{x^{2} - 16}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

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

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{4 x (x - 4)}{(x + 4) (x - 4)}$

Step 6

Cancel the common factor of $x - 4$ .

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Step 6.1

Cancel the common factor.

$\frac{4 x (x - 4)}{(x + 4) (x - 4)}$

Step 6.2

Rewrite the expression.

$\frac{4 x}{x + 4}$