Algebra Examples

$\frac{\frac{x}{9} + \frac{1}{3}}{1 - \frac{x^{2}}{9}}$

Step 1

Multiply the numerator and denominator of the fraction by $9$ .

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Step 1.1

Multiply $\frac{\frac{x}{9} + \frac{1}{3}}{1 - \frac{x^{2}}{9}}$ by $\frac{9}{9}$ .

$\frac{9}{9} \cdot \frac{\frac{x}{9} + \frac{1}{3}}{1 - \frac{x^{2}}{9}}$

Step 1.2

Combine.

$\frac{9 (\frac{x}{9} + \frac{1}{3})}{9 (1 - \frac{x^{2}}{9})}$

Step 2

Apply the distributive property.

$\frac{9 \frac{x}{9} + 9 (\frac{1}{3})}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $9$ .

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Step 3.1.1

Cancel the common factor.

$\frac{9 \frac{x}{9} + 9 (\frac{1}{3})}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3.1.2

Rewrite the expression.

$\frac{x + 9 (\frac{1}{3})}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3.2

Cancel the common factor of $3$ .

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Step 3.2.1

Factor $3$ out of $9$ .

$\frac{x + 3 (3) \frac{1}{3}}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3.2.2

Cancel the common factor.

$\frac{x + 3 \cdot 3 \frac{1}{3}}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3.2.3

Rewrite the expression.

$\frac{x + 3}{9 \cdot 1 + 9 (- \frac{x^{2}}{9})}$

Step 3.3

Cancel the common factor of $9$ .

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Step 3.3.1

Move the leading negative in $- \frac{x^{2}}{9}$ into the numerator.

$\frac{x + 3}{9 \cdot 1 + 9 \frac{- x^{2}}{9}}$

Step 3.3.2

Cancel the common factor.

$\frac{x + 3}{9 \cdot 1 + 9 \frac{- x^{2}}{9}}$

Step 3.3.3

Rewrite the expression.

$\frac{x + 3}{9 \cdot 1 - x^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9 \cdot 1$ as ${(3 \cdot 1)}^{2}$ .

$\frac{x + 3}{{(3 \cdot 1)}^{2} - x^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 \cdot 1$ and $b = x$ .

$\frac{x + 3}{(3 \cdot 1 + x) (3 \cdot 1 - x)}$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $3$ by $1$ .

$\frac{x + 3}{(3 + x) (3 \cdot 1 - x)}$

Step 4.3.2

Multiply $3$ by $1$ .

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

Step 5

Cancel the common factor of $x + 3$ and $3 + x$ .

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Step 5.1

Reorder terms.

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

$\frac{1}{3 - x}$