Algebra Examples

$\frac{{(a + 3)}^{2}}{a - 3} \div \frac{5}{(a^{2} - 9)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{{(a + 3)}^{2}}{a - 3} \cdot \frac{a^{2} - 9}{5}$

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 3$ .

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

Step 3

Cancel the common factor of $a - 3$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Multiply ${(a + 3)}^{2}$ by $a + 3$ by adding the exponents.

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Step 5.1

Multiply ${(a + 3)}^{2}$ by $a + 3$ .

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Step 5.1.1

Raise $a + 3$ to the power of $1$ .

$\frac{{(a + 3)}^{2} {(a + 3)}^{1}}{5}$

Step 5.1.2

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

$\frac{{(a + 3)}^{2 + 1}}{5}$

Step 5.2

Add $2$ and $1$ .

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

Step 6

Use the Binomial Theorem.

$\frac{a^{3} + 3 a^{2} \cdot 3 + 3 a \cdot 3^{2} + 3^{3}}{5}$

Step 7

Simplify each term.

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Step 7.1

Multiply $3$ by $3$ .

$\frac{a^{3} + 9 a^{2} + 3 a \cdot 3^{2} + 3^{3}}{5}$

Step 7.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 7.2.1

Move $3^{2}$ .

$\frac{a^{3} + 9 a^{2} + 3^{2} \cdot 3 a + 3^{3}}{5}$

Step 7.2.2

Multiply $3^{2}$ by $3$ .

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Step 7.2.2.1

Raise $3$ to the power of $1$ .

$\frac{a^{3} + 9 a^{2} + 3^{2} \cdot 3^{1} a + 3^{3}}{5}$

Step 7.2.2.2

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

$\frac{a^{3} + 9 a^{2} + 3^{2 + 1} a + 3^{3}}{5}$

Step 7.2.3

Add $2$ and $1$ .

$\frac{a^{3} + 9 a^{2} + 3^{3} a + 3^{3}}{5}$

Step 7.3

Raise $3$ to the power of $3$ .

$\frac{a^{3} + 9 a^{2} + 27 a + 3^{3}}{5}$

Step 7.4

Raise $3$ to the power of $3$ .

$\frac{a^{3} + 9 a^{2} + 27 a + 27}{5}$

Step 8

Split the fraction $\frac{a^{3} + 9 a^{2} + 27 a + 27}{5}$ into two fractions.

$\frac{a^{3} + 9 a^{2} + 27 a}{5} + \frac{27}{5}$

Step 9

Split the fraction $\frac{a^{3} + 9 a^{2} + 27 a}{5}$ into two fractions.

$\frac{a^{3} + 9 a^{2}}{5} + \frac{27 a}{5} + \frac{27}{5}$

Step 10

Split the fraction $\frac{a^{3} + 9 a^{2}}{5}$ into two fractions.

$\frac{a^{3}}{5} + \frac{9 a^{2}}{5} + \frac{27 a}{5} + \frac{27}{5}$