Basic Math Examples

$\frac{\frac{15 z^{2} + 25}{z - 4}}{6 z^{2} + \frac{10}{z - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{15 z^{2} + 25}{z - 4} \cdot \frac{1}{6 z^{2} + \frac{10}{z - 4}}$

Step 2

Factor $5$ out of $15 z^{2} + 25$ .

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Step 2.1

Factor $5$ out of $15 z^{2}$ .

$\frac{5 (3 z^{2}) + 25}{z - 4} \cdot \frac{1}{6 z^{2} + \frac{10}{z - 4}}$

Step 2.2

Factor $5$ out of $25$ .

$\frac{5 (3 z^{2}) + 5 (5)}{z - 4} \cdot \frac{1}{6 z^{2} + \frac{10}{z - 4}}$

Step 2.3

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

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{1}{6 z^{2} + \frac{10}{z - 4}}$

Step 3

Simplify the denominator.

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Step 3.1

To write $6 z^{2}$ as a fraction with a common denominator, multiply by $\frac{z - 4}{z - 4}$ .

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{1}{\frac{6 z^{2} (z - 4)}{z - 4} + \frac{10}{z - 4}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{1}{\frac{6 z^{2} (z - 4) + 10}{z - 4}}$

Step 3.3

Rewrite $\frac{6 z^{2} (z - 4) + 10}{z - 4}$ in a factored form.

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Step 3.3.1

Factor $2$ out of $6 z^{2} (z - 4) + 10$ .

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Step 3.3.1.1

Factor $2$ out of $6 z^{2} (z - 4)$ .

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{1}{\frac{2 (3 z^{2} (z - 4)) + 10}{z - 4}}$

Step 3.3.1.2

Factor $2$ out of $10$ .

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

Step 3.3.1.3

Factor $2$ out of $2 (3 z^{2} (z - 4)) + 2 (5)$ .

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

Step 3.3.2

Apply the distributive property.

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

Step 3.3.3

Multiply $z^{2}$ by $z$ by adding the exponents.

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Step 3.3.3.1

Move $z$ .

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

Step 3.3.3.2

Multiply $z$ by $z^{2}$ .

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Step 3.3.3.2.1

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

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

Step 3.3.3.2.2

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

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

Step 3.3.3.3

Add $1$ and $2$ .

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

Step 3.3.4

Multiply $- 4$ by $3$ .

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{1}{\frac{2 (3 z^{3} - 12 z^{2} + 5)}{z - 4}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 (3 z^{2} + 5)}{z - 4} (1 \frac{z - 4}{2 (3 z^{3} - 12 z^{2} + 5)})$

Step 5

Multiply $\frac{z - 4}{2 (3 z^{3} - 12 z^{2} + 5)}$ by $1$ .

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{z - 4}{2 (3 z^{3} - 12 z^{2} + 5)}$

Step 6

Cancel the common factor of $z - 4$ .

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Step 6.1

Cancel the common factor.

$\frac{5 (3 z^{2} + 5)}{z - 4} \cdot \frac{z - 4}{2 (3 z^{3} - 12 z^{2} + 5)}$

Step 6.2

Rewrite the expression.

$5 (3 z^{2} + 5) \frac{1}{2 (3 z^{3} - 12 z^{2} + 5)}$

Step 7

Combine $\frac{1}{2 (3 z^{3} - 12 z^{2} + 5)}$ and $5$ .

$\frac{5}{2 (3 z^{3} - 12 z^{2} + 5)} (3 z^{2} + 5)$

Step 8

Multiply $\frac{5}{2 (3 z^{3} - 12 z^{2} + 5)}$ by $3 z^{2} + 5$ .

$\frac{5 (3 z^{2} + 5)}{2 (3 z^{3} - 12 z^{2} + 5)}$