Precalculus Examples

$\frac{\frac{7}{81 x^{2} - 16} + x}{9 - \frac{5}{9 x + 4}}$

Step 1

Multiply the numerator and denominator of the fraction by $(81 x^{2} - 16) (9 x + 4)$ .

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Step 1.1

Multiply $\frac{\frac{7}{81 x^{2} - 16} + x}{9 - \frac{5}{9 x + 4}}$ by $\frac{(81 x^{2} - 16) (9 x + 4)}{(81 x^{2} - 16) (9 x + 4)}$ .

$\frac{(81 x^{2} - 16) (9 x + 4)}{(81 x^{2} - 16) (9 x + 4)} \cdot \frac{\frac{7}{81 x^{2} - 16} + x}{9 - \frac{5}{9 x + 4}}$

Step 1.2

Combine.

$\frac{(81 x^{2} - 16) (9 x + 4) (\frac{7}{81 x^{2} - 16} + x)}{(81 x^{2} - 16) (9 x + 4) (9 - \frac{5}{9 x + 4})}$

Step 2

Apply the distributive property.

$\frac{(81 x^{2} - 16) (9 x + 4) \frac{7}{81 x^{2} - 16} + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) (9 x + 4) (- \frac{5}{9 x + 4})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $81 x^{2} - 16$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(81 x^{2} - 16) (9 x + 4) \frac{7}{81 x^{2} - 16} + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) (9 x + 4) (- \frac{5}{9 x + 4})}$

Step 3.1.2

Rewrite the expression.

$\frac{(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) (9 x + 4) (- \frac{5}{9 x + 4})}$

Step 3.2

Cancel the common factor of $9 x + 4$ .

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Step 3.2.1

Move the leading negative in $- \frac{5}{9 x + 4}$ into the numerator.

$\frac{(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) (9 x + 4) \frac{- 5}{9 x + 4}}$

Step 3.2.2

Factor $9 x + 4$ out of $(81 x^{2} - 16) (9 x + 4)$ .

$\frac{(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (9 x + 4) (81 x^{2} - 16) \frac{- 5}{9 x + 4}}$

Step 3.2.3

Cancel the common factor.

$\frac{(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (9 x + 4) (81 x^{2} - 16) \frac{- 5}{9 x + 4}}$

Step 3.2.4

Rewrite the expression.

$\frac{(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $9 x + 4$ out of $(9 x + 4) \cdot 7 + (81 x^{2} - 16) (9 x + 4) x$ .

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Step 4.1.1

Factor $9 x + 4$ out of $(9 x + 4) \cdot 7$ .

$\frac{(9 x + 4) (7) + (81 x^{2} - 16) (9 x + 4) x}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.1.2

Factor $9 x + 4$ out of $(81 x^{2} - 16) (9 x + 4) x$ .

$\frac{(9 x + 4) (7) + (9 x + 4) ((81 x^{2} - 16) x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.1.3

Factor $9 x + 4$ out of $(9 x + 4) (7) + (9 x + 4) ((81 x^{2} - 16) x)$ .

$\frac{(9 x + 4) (7 + (81 x^{2} - 16) x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.2

Apply the distributive property.

$\frac{(9 x + 4) (7 + 81 x^{2} x - 16 x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.3

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

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Step 4.3.1

Move $x$ .

$\frac{(9 x + 4) (7 + 81 (x \cdot x^{2}) - 16 x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.3.2

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

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Step 4.3.2.1

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

$\frac{(9 x + 4) (7 + 81 (x^{1} x^{2}) - 16 x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.3.2.2

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

$\frac{(9 x + 4) (7 + 81 x^{1 + 2} - 16 x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.3.3

Add $1$ and $2$ .

$\frac{(9 x + 4) (7 + 81 x^{3} - 16 x)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 4.4

Reorder terms.

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $81 x^{2} - 16$ out of $(81 x^{2} - 16) (9 x + 4) \cdot 9 + (81 x^{2} - 16) \cdot - 5$ .

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Step 5.1.1

Factor $81 x^{2} - 16$ out of $(81 x^{2} - 16) (9 x + 4) \cdot 9$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(81 x^{2} - 16) ((9 x + 4) \cdot 9) + (81 x^{2} - 16) \cdot - 5}$

Step 5.1.2

Factor $81 x^{2} - 16$ out of $(81 x^{2} - 16) \cdot - 5$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(81 x^{2} - 16) ((9 x + 4) \cdot 9) + (81 x^{2} - 16) (- 5)}$

Step 5.1.3

Factor $81 x^{2} - 16$ out of $(81 x^{2} - 16) ((9 x + 4) \cdot 9) + (81 x^{2} - 16) (- 5)$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(81 x^{2} - 16) ((9 x + 4) \cdot 9 - 5)}$

Step 5.2

Rewrite $81 x^{2}$ as ${(9 x)}^{2}$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{({(9 x)}^{2} - 16) ((9 x + 4) \cdot 9 - 5)}$

Step 5.3

Rewrite $16$ as $4^{2}$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{({(9 x)}^{2} - 4^{2}) ((9 x + 4) \cdot 9 - 5)}$

Step 5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9 x$ and $b = 4$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) ((9 x + 4) \cdot 9 - 5)}$

Step 5.5

Apply the distributive property.

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) (9 x \cdot 9 + 4 \cdot 9 - 5)}$

Step 5.6

Multiply $9$ by $9$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) (81 x + 4 \cdot 9 - 5)}$

Step 5.7

Multiply $4$ by $9$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) (81 x + 36 - 5)}$

Step 5.8

Subtract $5$ from $36$ .

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) (81 x + 31)}$

Step 6

Cancel the common factor of $9 x + 4$ .

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Step 6.1

Cancel the common factor.

$\frac{(9 x + 4) (81 x^{3} - 16 x + 7)}{(9 x + 4) (9 x - 4) (81 x + 31)}$

Step 6.2

Rewrite the expression.

$\frac{81 x^{3} - 16 x + 7}{(9 x - 4) (81 x + 31)}$