Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

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 $b = 2$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Cancel the common factor of $x + 2$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 4

Factor using the perfect square rule.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Rewrite the polynomial.

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

Step 4.4

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

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

Step 5

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

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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