Algebra Examples

$\frac{1 - \frac{4}{x + 8}}{\frac{16}{x + 8} + x}$

Step 1

Multiply the numerator and denominator of the fraction by $x + 8$ .

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Step 1.1

Multiply $\frac{1 - \frac{4}{x + 8}}{\frac{16}{x + 8} + x}$ by $\frac{x + 8}{x + 8}$ .

$\frac{x + 8}{x + 8} \cdot \frac{1 - \frac{4}{x + 8}}{\frac{16}{x + 8} + x}$

Step 1.2

Combine.

$\frac{(x + 8) (1 - \frac{4}{x + 8})}{(x + 8) (\frac{16}{x + 8} + x)}$

Step 2

Apply the distributive property.

$\frac{(x + 8) \cdot 1 + (x + 8) (- \frac{4}{x + 8})}{(x + 8) \frac{16}{x + 8} + (x + 8) x}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 8$ .

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Step 3.1.1

Move the leading negative in $- \frac{4}{x + 8}$ into the numerator.

$\frac{(x + 8) \cdot 1 + (x + 8) \frac{- 4}{x + 8}}{(x + 8) \frac{16}{x + 8} + (x + 8) x}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 8) \cdot 1 + (x + 8) \frac{- 4}{x + 8}}{(x + 8) \frac{16}{x + 8} + (x + 8) x}$

Step 3.1.3

Rewrite the expression.

$\frac{(x + 8) \cdot 1 - 4}{(x + 8) \frac{16}{x + 8} + (x + 8) x}$

Step 3.2

Cancel the common factor of $x + 8$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(x + 8) \cdot 1 - 4}{(x + 8) \frac{16}{x + 8} + (x + 8) x}$

Step 3.2.2

Rewrite the expression.

$\frac{(x + 8) \cdot 1 - 4}{16 + (x + 8) x}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x + 8$ by $1$ .

$\frac{x + 8 - 4}{16 + (x + 8) x}$

Step 4.2

Subtract $4$ from $8$ .

$\frac{x + 4}{16 + (x + 8) x}$

Step 5

Simplify the denominator.

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Step 5.1

Apply the distributive property.

$\frac{x + 4}{16 + x \cdot x + 8 x}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x + 4}{16 + x^{2} + 8 x}$

Step 5.3

Reorder terms.

$\frac{x + 4}{x^{2} + 8 x + 16}$

Step 5.4

Factor using the perfect square rule.

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Step 5.4.1

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

$\frac{x + 4}{x^{2} + 8 x + 4^{2}}$

Step 5.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 5.4.3

Rewrite the polynomial.

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

Step 5.4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 6

Cancel the common factor of $x + 4$ and ${(x + 4)}^{2}$ .

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Step 6.1

Multiply by $1$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

$\frac{1}{x + 4}$