Basic Math Examples

$\frac{5 z + 9}{z^{2} - 16} \cdot \frac{3 z}{z^{2} - 8 z + 16}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{5 z + 9}{z^{2} - 4^{2}} \cdot \frac{3 z}{z^{2} - 8 z + 16}$

Step 1.2

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

$\frac{5 z + 9}{(z + 4) (z - 4)} \cdot \frac{3 z}{z^{2} - 8 z + 16}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{5 z + 9}{(z + 4) (z - 4)} \cdot \frac{3 z}{z^{2} - 8 z + 4^{2}}$

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.

$8 z = 2 \cdot z \cdot 4$

Step 2.3

Rewrite the polynomial.

$\frac{5 z + 9}{(z + 4) (z - 4)} \cdot \frac{3 z}{z^{2} - 2 \cdot z \cdot 4 + 4^{2}}$

Step 2.4

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

$\frac{5 z + 9}{(z + 4) (z - 4)} \cdot \frac{3 z}{{(z - 4)}^{2}}$

Step 3

Multiply $\frac{5 z + 9}{(z + 4) (z - 4)} \cdot \frac{3 z}{{(z - 4)}^{2}}$ .

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Step 3.1

Multiply $\frac{5 z + 9}{(z + 4) (z - 4)}$ by $\frac{3 z}{{(z - 4)}^{2}}$ .

$\frac{(5 z + 9) (3 z)}{(z + 4) (z - 4) {(z - 4)}^{2}}$

Step 3.2

Multiply $z - 4$ by ${(z - 4)}^{2}$ by adding the exponents.

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Step 3.2.1

Move ${(z - 4)}^{2}$ .

$\frac{(5 z + 9) (3 z)}{(z + 4) ({(z - 4)}^{2} (z - 4))}$

Step 3.2.2

Multiply ${(z - 4)}^{2}$ by $z - 4$ .

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Step 3.2.2.1

Raise $z - 4$ to the power of $1$ .

$\frac{(5 z + 9) (3 z)}{(z + 4) ({(z - 4)}^{2} {(z - 4)}^{1})}$

Step 3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(5 z + 9) (3 z)}{(z + 4) {(z - 4)}^{2 + 1}}$

Step 3.2.3

Add $2$ and $1$ .

$\frac{(5 z + 9) (3 z)}{(z + 4) {(z - 4)}^{3}}$

Step 4

Move $3$ to the left of $5 z + 9$ .

$\frac{3 \cdot (5 z + 9) z}{(z + 4) {(z - 4)}^{3}}$

Step 5

Reorder factors in $\frac{3 (5 z + 9) z}{(z + 4) {(z - 4)}^{3}}$ .

$\frac{3 z (5 z + 9)}{(z + 4) {(z - 4)}^{3}}$