Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify terms.

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Step 2.1

Factor $4$ out of $4 u + 8$ .

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Step 2.1.1

Factor $4$ out of $4 u$ .

$\frac{u^{2}}{4 (u) + 8} \cdot \frac{8 u + 16}{2 u^{2} - 8 u + 8}$

Step 2.1.2

Factor $4$ out of $8$ .

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

Step 2.1.3

Factor $4$ out of $4 u + 4 \cdot 2$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{8 u + 16}{2 u^{2} - 8 u + 8}$

Step 2.2

Cancel the common factor of $8 u + 16$ and $2 u^{2} - 8 u + 8$ .

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Step 2.2.1

Factor $2$ out of $8 u$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u) + 16}{2 u^{2} - 8 u + 8}$

Step 2.2.2

Factor $2$ out of $16$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u) + 2 \cdot 8}{2 u^{2} - 8 u + 8}$

Step 2.2.3

Factor $2$ out of $2 (4 u) + 2 (8)$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 u^{2} - 8 u + 8}$

Step 2.2.4

Cancel the common factors.

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Step 2.2.4.1

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

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2}) - 8 u + 8}$

Step 2.2.4.2

Factor $2$ out of $- 8 u$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2}) + 2 (- 4 u) + 8}$

Step 2.2.4.3

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

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2} - 4 u) + 8}$

Step 2.2.4.4

Factor $2$ out of $8$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2} - 4 u) + 2 \cdot 4}$

Step 2.2.4.5

Factor $2$ out of $2 (u^{2} - 4 u) + 2 \cdot 4$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2} - 4 u + 4)}$

Step 2.2.4.6

Cancel the common factor.

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{2 (4 u + 8)}{2 (u^{2} - 4 u + 4)}$

Step 2.2.4.7

Rewrite the expression.

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{4 u + 8}{u^{2} - 4 u + 4}$

Step 2.3

Factor $4$ out of $4 u + 8$ .

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Step 2.3.1

Factor $4$ out of $4 u$ .

$\frac{u^{2}}{4 (u + 2)} \cdot \frac{4 (u) + 8}{u^{2} - 4 u + 4}$

Step 2.3.2

Factor $4$ out of $8$ .

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

Step 2.3.3

Factor $4$ out of $4 u + 4 \cdot 2$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

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

$4 u = 2 \cdot u \cdot 2$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $4$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\frac{u^{2} (u + 2)}{(u + 2) {(u - 2)}^{2}}$

Step 4.3

Cancel the common factor of $u + 2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{u^{2} (u + 2)}{(u + 2) {(u - 2)}^{2}}$

Step 4.3.2

Rewrite the expression.

$\frac{u^{2}}{{(u - 2)}^{2}}$