Basic Math Examples

$\frac{8 b + 24}{3 a + 12} \div \frac{a b - 4 b + 3 a - 12}{a^{2} - 8 a + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{8 b + 24}{3 a + 12} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor

$8$ out of

$8 b + 24$ .

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Step 2.1.1

Factor

$8$ out of

$8 b$ .

$\frac{8 (b) + 24}{3 a + 12} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2.1.2

Factor

$8$ out of

$24$ .

$\frac{8 b + 8 \cdot 3}{3 a + 12} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2.1.3

Factor

$8$ out of

$8 b + 8 \cdot 3$ .

$\frac{8 (b + 3)}{3 a + 12} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2.2

Factor

$3$ out of

$3 a + 12$ .

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Step 2.2.1

Factor

$3$ out of

$3 a$ .

$\frac{8 (b + 3)}{3 (a) + 12} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2.2.2

Factor

$3$ out of

$12$ .

$\frac{8 (b + 3)}{3 a + 3 \cdot 4} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 2.2.3

Factor

$3$ out of

$3 a + 3 \cdot 4$ .

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{a^{2} - 8 a + 16}{a b - 4 b + 3 a - 12}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite

$16$ as

$4^{2}$ .

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{a^{2} - 8 a + 4^{2}}{a b - 4 b + 3 a - 12}$

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.

$8 a = 2 \cdot a \cdot 4$

Step 3.3

Rewrite the polynomial.

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{a^{2} - 2 \cdot a \cdot 4 + 4^{2}}{a b - 4 b + 3 a - 12}$

Step 3.4

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = a$ and

$b = 4$ .

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{{(a - 4)}^{2}}{a b - 4 b + 3 a - 12}$

Step 4

Simplify the denominator.

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Step 4.1

Factor out the greatest common factor from each group.

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Step 4.1.1

Group the first two terms and the last two terms.

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{{(a - 4)}^{2}}{(a b - 4 b) + 3 a - 12}$

Step 4.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{{(a - 4)}^{2}}{b (a - 4) + 3 (a - 4)}$

Step 4.2

Factor the polynomial by factoring out the greatest common factor,

$a - 4$ .

$\frac{8 (b + 3)}{3 (a + 4)} \cdot \frac{{(a - 4)}^{2}}{(a - 4) (b + 3)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{8 (b + 3) {(a - 4)}^{2}}{3 (a + 4) ((a - 4) (b + 3))}$

Step 5.2

Cancel the common factor of

$b + 3$ .

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Step 5.2.1

Cancel the common factor.

$\frac{8 (b + 3) {(a - 4)}^{2}}{3 (a + 4) ((a - 4) (b + 3))}$

Step 5.2.2

Rewrite the expression.

$\frac{8 {(a - 4)}^{2}}{3 (a + 4) (a - 4)}$

Step 5.3

Cancel the common factor of

${(a - 4)}^{2}$ and

$a - 4$ .

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Step 5.3.1

Factor

$a - 4$ out of

$8 {(a - 4)}^{2}$ .

$\frac{(a - 4) (8 (a - 4))}{3 (a + 4) (a - 4)}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor

$a - 4$ out of

$3 (a + 4) (a - 4)$ .

$\frac{(a - 4) (8 (a - 4))}{(a - 4) (3 (a + 4))}$

Step 5.3.2.2

Cancel the common factor.

$\frac{(a - 4) (8 (a - 4))}{(a - 4) (3 (a + 4))}$

Step 5.3.2.3

Rewrite the expression.

$\frac{8 (a - 4)}{3 (a + 4)}$