Algebra Examples

$\frac{10 a^{4}}{a^{2} - 9} \cdot \frac{a^{2} - 3 a}{15 a^{2}}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{10 a^{4}}{a^{2} - 3^{2}} \cdot \frac{a^{2} - 3 a}{15 a^{2}}$

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 $a = a$ and $b = 3$ .

$\frac{10 a^{4}}{(a + 3) (a - 3)} \cdot \frac{a^{2} - 3 a}{15 a^{2}}$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

$\frac{10 a^{4}}{(a + 3) (a - 3)} \cdot \frac{a \cdot a - 3 a}{15 a^{2}}$

Step 2.1.2

Factor $a$ out of $- 3 a$ .

$\frac{10 a^{4}}{(a + 3) (a - 3)} \cdot \frac{a \cdot a + a \cdot - 3}{15 a^{2}}$

Step 2.1.3

Factor $a$ out of $a \cdot a + a \cdot - 3$ .

$\frac{10 a^{4}}{(a + 3) (a - 3)} \cdot \frac{a (a - 3)}{15 a^{2}}$

Step 2.2

Combine.

$\frac{10 a^{4} (a (a - 3))}{(a + 3) (a - 3) (15 a^{2})}$

Step 3

Multiply $a^{4}$ by $a$ by adding the exponents.

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Step 3.1

Move $a$ .

$\frac{10 (a \cdot a^{4}) (a - 3)}{(a + 3) (a - 3) (15 a^{2})}$

Step 3.2

Multiply $a$ by $a^{4}$ .

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Step 3.2.1

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

$\frac{10 (a^{1} a^{4}) (a - 3)}{(a + 3) (a - 3) (15 a^{2})}$

Step 3.2.2

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

$\frac{10 a^{1 + 4} (a - 3)}{(a + 3) (a - 3) (15 a^{2})}$

Step 3.3

Add $1$ and $4$ .

$\frac{10 a^{5} (a - 3)}{(a + 3) (a - 3) (15 a^{2})}$

Step 4

Cancel the common factor of $10$ and $15$ .

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Step 4.1

Factor $5$ out of $10 a^{5} (a - 3)$ .

$\frac{5 (2 a^{5} (a - 3))}{(a + 3) (a - 3) (15 a^{2})}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $5$ out of $(a + 3) (a - 3) (15 a^{2})$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $a^{5}$ and $a^{2}$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

Cancel the common factor of $a - 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{2 a^{3}}{(a + 3) \cdot (3)}$

Step 7

Move $3$ to the left of $a + 3$ .

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