Basic Math Examples

$\frac{81 z^{2}}{5 z^{2} - 40 z + 80} \cdot \frac{5 z - 20}{9 z}$

Step 1

Simplify the denominator.

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Step 1.1

Factor $5$ out of $5 z^{2} - 40 z + 80$ .

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Step 1.1.1

Factor $5$ out of $5 z^{2}$ .

$\frac{81 z^{2}}{5 (z^{2}) - 40 z + 80} \cdot \frac{5 z - 20}{9 z}$

Step 1.1.2

Factor $5$ out of $- 40 z$ .

$\frac{81 z^{2}}{5 (z^{2}) + 5 (- 8 z) + 80} \cdot \frac{5 z - 20}{9 z}$

Step 1.1.3

Factor $5$ out of $80$ .

$\frac{81 z^{2}}{5 z^{2} + 5 (- 8 z) + 5 \cdot 16} \cdot \frac{5 z - 20}{9 z}$

Step 1.1.4

Factor $5$ out of $5 z^{2} + 5 (- 8 z)$ .

$\frac{81 z^{2}}{5 (z^{2} - 8 z) + 5 \cdot 16} \cdot \frac{5 z - 20}{9 z}$

Step 1.1.5

Factor $5$ out of $5 (z^{2} - 8 z) + 5 \cdot 16$ .

$\frac{81 z^{2}}{5 (z^{2} - 8 z + 16)} \cdot \frac{5 z - 20}{9 z}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

$\frac{81 z^{2}}{5 (z^{2} - 8 z + 4^{2})} \cdot \frac{5 z - 20}{9 z}$

Step 1.2.2

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 1.2.3

Rewrite the polynomial.

$\frac{81 z^{2}}{5 (z^{2} - 2 \cdot z \cdot 4 + 4^{2})} \cdot \frac{5 z - 20}{9 z}$

Step 1.2.4

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

$\frac{81 z^{2}}{5 {(z - 4)}^{2}} \cdot \frac{5 z - 20}{9 z}$

Step 2

Simplify terms.

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Step 2.1

Factor $5$ out of $5 z - 20$ .

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Step 2.1.1

Factor $5$ out of $5 z$ .

$\frac{81 z^{2}}{5 {(z - 4)}^{2}} \cdot \frac{5 (z) - 20}{9 z}$

Step 2.1.2

Factor $5$ out of $- 20$ .

$\frac{81 z^{2}}{5 {(z - 4)}^{2}} \cdot \frac{5 z + 5 \cdot - 4}{9 z}$

Step 2.1.3

Factor $5$ out of $5 z + 5 \cdot - 4$ .

$\frac{81 z^{2}}{5 {(z - 4)}^{2}} \cdot \frac{5 (z - 4)}{9 z}$

Step 2.2

Combine.

$\frac{81 z^{2} (5 (z - 4))}{5 {(z - 4)}^{2} (9 z)}$

Step 2.3

Cancel the common factor of $81$ and $9$ .

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Step 2.3.1

Factor $9$ out of $81 z^{2} (5 (z - 4))$ .

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $9$ out of $5 {(z - 4)}^{2} (9 z)$ .

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 2.4

Cancel the common factor of $z^{2}$ and $z$ .

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Step 2.4.1

Factor $z$ out of $9 z^{2} (5 (z - 4))$ .

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

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $z$ out of $5 {(z - 4)}^{2} (z)$ .

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

Step 2.4.2.2

Cancel the common factor.

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

Step 2.4.2.3

Rewrite the expression.

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

Step 2.5

Cancel the common factor of $5$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

Cancel the common factor of $z - 4$ and ${(z - 4)}^{2}$ .

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Step 2.6.1

Factor $z - 4$ out of $9 z (z - 4)$ .

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

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

$\frac{(z - 4) (9 z)}{(z - 4) (z - 4)}$

Step 2.6.2.2

Cancel the common factor.

$\frac{(z - 4) (9 z)}{(z - 4) (z - 4)}$

Step 2.6.2.3

Rewrite the expression.

$\frac{9 z}{z - 4}$