Pre-Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x - 3$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

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

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

$(x + 9) \cdot 1$

Step 2.3

Multiply $x + 9$ by $1$ .

$x + 9$