Basic Math Examples

$\frac{5 a^{2} - 45}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $5$ out of $5 a^{2} - 45$ .

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Step 1.1.1

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

$\frac{5 (a^{2}) - 45}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 1.1.2

Factor $5$ out of $- 45$ .

$\frac{5 a^{2} + 5 \cdot - 9}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 1.1.3

Factor $5$ out of $5 a^{2} + 5 \cdot - 9$ .

$\frac{5 (a^{2} - 9)}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 1.2

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

$\frac{5 (a^{2} - 3^{2})}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$\frac{5 (a + 3) (a - 3)}{4 a^{2} - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2

Simplify terms.

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Step 2.1

Factor $4 a$ out of $4 a^{2} - 20 a$ .

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Step 2.1.1

Factor $4 a$ out of $4 a^{2}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a) - 20 a} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.1.2

Factor $4 a$ out of $- 20 a$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a) + 4 a (- 5)} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.1.3

Factor $4 a$ out of $4 a (a) + 4 a (- 5)$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{3} + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.2

Factor $a^{2}$ out of $a^{3} + 3 a^{2}$ .

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Step 2.2.1

Factor $a^{2}$ out of $a^{3}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} a + 3 a^{2}}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.2.2

Factor $a^{2}$ out of $3 a^{2}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} a + a^{2} \cdot 3}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.2.3

Factor $a^{2}$ out of $a^{2} a + a^{2} \cdot 3$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} (a + 3)}{2 a^{2} - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.3

Factor $2 a$ out of $2 a^{2} - 10 a$ .

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Step 2.3.1

Factor $2 a$ out of $2 a^{2}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} (a + 3)}{2 a (a) - 10 a} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.3.2

Factor $2 a$ out of $- 10 a$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} (a + 3)}{2 a (a) + 2 a (- 5)} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.3.3

Factor $2 a$ out of $2 a (a) + 2 a (- 5)$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a^{2} (a + 3)}{2 a (a - 5)} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.4

Reduce the expression $\frac{a^{2} (a + 3)}{2 a (a - 5)}$ by cancelling the common factors.

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Step 2.4.1

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

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a (a + 3))}{2 a (a - 5)} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.4.2

Factor $a$ out of $2 a (a - 5)$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a (a + 3))}{a (2 (a - 5))} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.4.3

Cancel the common factor.

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a (a + 3))}{a (2 (a - 5))} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.4.4

Rewrite the expression.

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{12 a^{3} + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.5

Factor $4 a^{2}$ out of $12 a^{3} + 16 a^{2}$ .

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Step 2.5.1

Factor $4 a^{2}$ out of $12 a^{3}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a) + 16 a^{2}}{2 a^{2} - 6 a}$

Step 2.5.2

Factor $4 a^{2}$ out of $16 a^{2}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a) + 4 a^{2} (4)}{2 a^{2} - 6 a}$

Step 2.5.3

Factor $4 a^{2}$ out of $4 a^{2} (3 a) + 4 a^{2} (4)$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a + 4)}{2 a^{2} - 6 a}$

Step 2.6

Factor $2 a$ out of $2 a^{2} - 6 a$ .

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Step 2.6.1

Factor $2 a$ out of $2 a^{2}$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a + 4)}{2 a (a) - 6 a}$

Step 2.6.2

Factor $2 a$ out of $- 6 a$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a + 4)}{2 a (a) + 2 a (- 3)}$

Step 2.6.3

Factor $2 a$ out of $2 a (a) + 2 a (- 3)$ .

$\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} \div \frac{a (a + 3)}{2 (a - 5)} \cdot \frac{4 a^{2} (3 a + 4)}{2 a (a - 3)}$

Step 2.7

Combine.

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

Step 2.8

Cancel the common factor of $4$ and $2$ .

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Step 2.8.1

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

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

Step 2.8.2

Cancel the common factors.

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Step 2.8.2.1

Factor $2$ out of $\frac{a (a + 3)}{2 (a - 5)} (2 a (a - 3))$ .

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

Step 2.8.2.2

Cancel the common factor.

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

Step 2.8.2.3

Rewrite the expression.

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

Step 2.9

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

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Step 2.9.1

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

$\frac{a (\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 a (3 a + 4)))}{\frac{a (a + 3)}{2 (a - 5)} (a (a - 3))}$

Step 2.9.2

Cancel the common factors.

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Step 2.9.2.1

Factor $a$ out of $\frac{a (a + 3)}{2 (a - 5)} (a (a - 3))$ .

$\frac{a (\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 a (3 a + 4)))}{a (\frac{a + 3}{2 (a - 5)} (a (a - 3)))}$

Step 2.9.2.2

Cancel the common factor.

$\frac{a (\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 a (3 a + 4)))}{a (\frac{a + 3}{2 (a - 5)} (a (a - 3)))}$

Step 2.9.2.3

Rewrite the expression.

$\frac{\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 a (3 a + 4))}{\frac{a + 3}{2 (a - 5)} (a (a - 3))}$

Step 2.10

Cancel the common factor of $a$ .

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Step 2.10.1

Cancel the common factor.

$\frac{\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 a (3 a + 4))}{\frac{a + 3}{2 (a - 5)} (a (a - 3))}$

Step 2.10.2

Rewrite the expression.

$\frac{\frac{5 (a + 3) (a - 3)}{4 a (a - 5)} (2 (3 a + 4))}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 2.11

Combine $2$ and $\frac{5 (a + 3) (a - 3)}{4 a (a - 5)}$ .

$\frac{\frac{2 (5 (a + 3) (a - 3))}{4 a (a - 5)} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 3

Multiply $2$ by $5$ .

$\frac{\frac{10 (a + 3) (a - 3)}{4 a (a - 5)} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4

Simplify terms.

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Step 4.1

Reduce the expression $\frac{10 (a + 3) (a - 3)}{4 a (a - 5)}$ by cancelling the common factors.

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Step 4.1.1

Factor $2$ out of $10 (a + 3) (a - 3)$ .

$\frac{\frac{2 (5 (a + 3) (a - 3))}{4 a (a - 5)} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.1.2

Factor $2$ out of $4 a (a - 5)$ .

$\frac{\frac{2 (5 (a + 3) (a - 3))}{2 (2 a (a - 5))} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.1.3

Cancel the common factor.

$\frac{\frac{2 (5 (a + 3) (a - 3))}{2 (2 a (a - 5))} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.1.4

Rewrite the expression.

$\frac{\frac{5 (a + 3) (a - 3)}{2 a (a - 5)} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.2

Factor $\frac{5 (a + 3) (a - 3)}{2 a (a - 5)}$ out of $\frac{\frac{5 (a + 3) (a - 3)}{2 a (a - 5)} (3 a + 4)}{\frac{a + 3}{2 (a - 5)} (a - 3)}$ .

$\frac{5 (a + 3) (a - 3)}{2 a (a - 5)} \cdot \frac{3 a + 4}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.3

Cancel the common factor of $a - 3$ .

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Step 4.3.1

Factor $a - 3$ out of $5 (a + 3) (a - 3)$ .

$\frac{(a - 3) (5 (a + 3))}{2 a (a - 5)} \cdot \frac{3 a + 4}{\frac{a + 3}{2 (a - 5)} (a - 3)}$

Step 4.3.2

Factor $a - 3$ out of $\frac{a + 3}{2 (a - 5)} (a - 3)$ .

$\frac{(a - 3) (5 (a + 3))}{2 a (a - 5)} \cdot \frac{3 a + 4}{(a - 3) \frac{a + 3}{2 (a - 5)}}$

Step 4.3.3

Cancel the common factor.

$\frac{(a - 3) (5 (a + 3))}{2 a (a - 5)} \cdot \frac{3 a + 4}{(a - 3) \frac{a + 3}{2 (a - 5)}}$

Step 4.3.4

Rewrite the expression.

$\frac{5 (a + 3)}{2 a (a - 5)} \cdot \frac{3 a + 4}{\frac{a + 3}{2 (a - 5)}}$

Step 4.4

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

$\frac{5 (a + 3) (3 a + 4)}{2 a (a - 5) \frac{a + 3}{2 (a - 5)}}$

Step 4.5

Combine $2$ and $\frac{a + 3}{2 (a - 5)}$ .

$\frac{5 (a + 3) (3 a + 4)}{a (a - 5) \frac{2 (a + 3)}{2 (a - 5)}}$

Step 4.6

Combine $\frac{2 (a + 3)}{2 (a - 5)}$ and $a$ .

$\frac{5 (a + 3) (3 a + 4)}{\frac{2 (a + 3) a}{2 (a - 5)} (a - 5)}$

Step 4.7

Cancel the common factor of $2$ .

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Step 4.7.1

Cancel the common factor.

$\frac{5 (a + 3) (3 a + 4)}{\frac{2 (a + 3) a}{2 (a - 5)} (a - 5)}$

Step 4.7.2

Rewrite the expression.

$\frac{5 (a + 3) (3 a + 4)}{\frac{(a + 3) a}{a - 5} (a - 5)}$

Step 5

Factor $\frac{(a + 3) a}{a - 5}$ out of $\frac{5 (a + 3) (3 a + 4)}{\frac{(a + 3) a}{a - 5} (a - 5)}$ .

$\frac{a - 5}{(a + 3) a} \cdot \frac{5 (a + 3) (3 a + 4)}{a - 5}$

Step 6

Cancel the common factor of $a - 5$ .

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Step 6.1

Cancel the common factor.

$\frac{a - 5}{(a + 3) a} \cdot \frac{5 (a + 3) (3 a + 4)}{a - 5}$

Step 6.2

Rewrite the expression.

$\frac{1}{(a + 3) a} (5 (a + 3) (3 a + 4))$

Step 7

Cancel the common factor of $a + 3$ .

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Step 7.1

Factor $a + 3$ out of $(a + 3) a$ .

$\frac{1}{(a + 3) (a)} (5 (a + 3) (3 a + 4))$

Step 7.2

Factor $a + 3$ out of $5 (a + 3) (3 a + 4)$ .

$\frac{1}{(a + 3) (a)} ((a + 3) (5 (3 a + 4)))$

Step 7.3

Cancel the common factor.

$\frac{1}{(a + 3) a} ((a + 3) (5 (3 a + 4)))$

Step 7.4

Rewrite the expression.

$\frac{1}{a} (5 (3 a + 4))$

Step 8

Combine $5$ and $\frac{1}{a}$ .

$\frac{5}{a} (3 a + 4)$

Step 9

Multiply $\frac{5}{a}$ by $3 a + 4$ .

$\frac{5 (3 a + 4)}{a}$