Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $27$ as $3^{3}$ .

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

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

Raise $3$ to the power of $2$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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 $b = 3$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $a^{2} + 3 a + 9$ .

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Step 4.1.1

Factor $a^{2} + 3 a + 9$ out of $(a - 3) (a^{2} + 3 a + 9)$ .

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $a + 3$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\frac{a - 3}{a - 3}$

Step 4.3

Cancel the common factor of $a - 3$ .

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Step 4.3.1

Cancel the common factor.

$\frac{a - 3}{a - 3}$

Step 4.3.2

Rewrite the expression.

$1$