Pre-Algebra Examples

$\frac{\frac{x^{4} - 16}{x^{3} + 27}}{\frac{x + 2}{2 x + 6}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{4} - 16}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{{(x^{2})}^{2} - 16}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 2.2

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

$\frac{{(x^{2})}^{2} - 4^{2}}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 2.3

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

$\frac{(x^{2} + 4) (x^{2} - 4)}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 2.4

Simplify.

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Step 2.4.1

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

$\frac{(x^{2} + 4) (x^{2} - 2^{2})}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 2.4.2

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 $b = 2$ .

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{x^{3} + 27} \cdot \frac{2 x + 6}{x + 2}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $27$ as $3^{3}$ .

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{x^{3} + 3^{3}} \cdot \frac{2 x + 6}{x + 2}$

Step 3.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 3$ .

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{(x + 3) (x^{2} - x \cdot 3 + 3^{2})} \cdot \frac{2 x + 6}{x + 2}$

Step 3.3

Simplify.

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Step 3.3.1

Multiply $3$ by $- 1$ .

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{(x + 3) (x^{2} - 3 x + 3^{2})} \cdot \frac{2 x + 6}{x + 2}$

Step 3.3.2

Raise $3$ to the power of $2$ .

$\frac{(x^{2} + 4) (x + 2) (x - 2)}{(x + 3) (x^{2} - 3 x + 9)} \cdot \frac{2 x + 6}{x + 2}$

Step 4

Cancel the common factor of $x + 2$ .

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Step 4.1

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

$\frac{(x + 2) ((x^{2} + 4) (x - 2))}{(x + 3) (x^{2} - 3 x + 9)} \cdot \frac{2 x + 6}{x + 2}$

Step 4.2

Cancel the common factor.

$\frac{(x + 2) ((x^{2} + 4) (x - 2))}{(x + 3) (x^{2} - 3 x + 9)} \cdot \frac{2 x + 6}{x + 2}$

Step 4.3

Rewrite the expression.

$\frac{(x^{2} + 4) (x - 2)}{(x + 3) (x^{2} - 3 x + 9)} (2 x + 6)$

Step 5

Multiply $\frac{(x^{2} + 4) (x - 2)}{(x + 3) (x^{2} - 3 x + 9)}$ by $2 x + 6$ .

$\frac{(x^{2} + 4) (x - 2) (2 x + 6)}{(x + 3) (x^{2} - 3 x + 9)}$

Step 6

Factor $2$ out of $2 x + 6$ .

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Step 6.1

Factor $2$ out of $2 x$ .

$\frac{(x^{2} + 4) (x - 2) (2 (x) + 6)}{(x + 3) (x^{2} - 3 x + 9)}$

Step 6.2

Factor $2$ out of $6$ .

$\frac{(x^{2} + 4) (x - 2) (2 x + 2 \cdot 3)}{(x + 3) (x^{2} - 3 x + 9)}$

Step 6.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

$\frac{(x^{2} + 4) (x - 2) (2 (x + 3))}{(x + 3) (x^{2} - 3 x + 9)}$

$\frac{(x^{2} + 4) (x - 2) \cdot 2 (x + 3)}{(x + 3) (x^{2} - 3 x + 9)}$

Step 7

Cancel the common factor of $x + 3$ .

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Step 7.1

Cancel the common factor.

$\frac{(x^{2} + 4) (x - 2) \cdot 2 (x + 3)}{(x + 3) (x^{2} - 3 x + 9)}$

Step 7.2

Rewrite the expression.

$\frac{(x^{2} + 4) (x - 2) \cdot 2}{x^{2} - 3 x + 9}$

Step 8

Move $2$ to the left of $(x^{2} + 4) (x - 2)$ .

$\frac{2 (x^{2} + 4) (x - 2)}{x^{2} - 3 x + 9}$