Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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 2.3

Rewrite the polynomial.

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

Step 2.4

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

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $x$ out of $4 x$ .

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

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 4$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Combine $x$ and $\frac{x + 4}{x + 4}$ .

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

Step 3.5

Cancel the common factor of $x + 4$ .

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Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

Divide $x$ by $1$ .

$x$