Basic Math Examples

$\frac{z^{2} - 9}{25 z + 75} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$\frac{z^{2} - 3^{2}}{25 z + 75} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

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 $b = 3$ .

$\frac{(z + 3) (z - 3)}{25 z + 75} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $25$ out of $25 z + 75$ .

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Step 2.1.1

Factor $25$ out of $25 z$ .

$\frac{(z + 3) (z - 3)}{25 (z) + 75} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

Step 2.1.2

Factor $25$ out of $75$ .

$\frac{(z + 3) (z - 3)}{25 z + 25 \cdot 3} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

Step 2.1.3

Factor $25$ out of $25 z + 25 \cdot 3$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 z + 375}{5 z^{2} - 30 z + 45}$

Step 2.2

Factor $125$ out of $125 z + 375$ .

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Step 2.2.1

Factor $125$ out of $125 z$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z) + 375}{5 z^{2} - 30 z + 45}$

Step 2.2.2

Factor $125$ out of $375$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 z + 125 \cdot 3}{5 z^{2} - 30 z + 45}$

Step 2.2.3

Factor $125$ out of $125 z + 125 \cdot 3$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 z^{2} - 30 z + 45}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $5$ out of $5 z^{2} - 30 z + 45$ .

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Step 3.1.1

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

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2}) - 30 z + 45}$

Step 3.1.2

Factor $5$ out of $- 30 z$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2}) + 5 (- 6 z) + 45}$

Step 3.1.3

Factor $5$ out of $45$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 z^{2} + 5 (- 6 z) + 5 \cdot 9}$

Step 3.1.4

Factor $5$ out of $5 z^{2} + 5 (- 6 z)$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2} - 6 z) + 5 \cdot 9}$

Step 3.1.5

Factor $5$ out of $5 (z^{2} - 6 z) + 5 \cdot 9$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2} - 6 z + 9)}$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2} - 6 z + 3^{2})}$

Step 3.2.2

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

$6 z = 2 \cdot z \cdot 3$

Step 3.2.3

Rewrite the polynomial.

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 (z^{2} - 2 \cdot z \cdot 3 + 3^{2})}$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 3$ .

$\frac{(z + 3) (z - 3)}{25 (z + 3)} \cdot \frac{125 (z + 3)}{5 {(z - 3)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{(z + 3) (z - 3) (125 (z + 3))}{25 (z + 3) (5 {(z - 3)}^{2})}$

Step 4.2

Cancel the common factor of $z + 3$ .

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Step 4.2.1

Cancel the common factor.

$\frac{(z + 3) (z - 3) (125 (z + 3))}{25 (z + 3) (5 {(z - 3)}^{2})}$

Step 4.2.2

Rewrite the expression.

$\frac{(z - 3) (125 (z + 3))}{25 (5 {(z - 3)}^{2})}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $z - 3$ out of $25 (5 {(z - 3)}^{2})$ .

$\frac{(z - 3) (125 (z + 3))}{(z - 3) (25 (5 (z - 3)))}$

Step 5.2

Cancel the common factor.

$\frac{(z - 3) (125 (z + 3))}{(z - 3) (25 (5 (z - 3)))}$

Step 5.3

Rewrite the expression.

$\frac{125 (z + 3)}{25 (5 (z - 3))}$

Step 6

Cancel the common factor of $125$ and $25$ .

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Step 6.1

Factor $25$ out of $125 (z + 3)$ .

$\frac{25 (5 (z + 3))}{25 (5 (z - 3))}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

$\frac{25 (5 (z + 3))}{25 (5 (z - 3))}$

Step 6.2.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $5$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{z + 3}{z - 3}$