Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Rewrite $x^{- 2}$ as ${(x^{- 1})}^{2}$ .

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.4.3

Combine the numerators over the common denominator.

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

Step 1.4.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.5

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.4.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $3 x - 1$ .

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Step 6.1

Factor $3 x - 1$ out of $(3 x + 1) (3 x - 1)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Factor $x$ out of $x^{2}$ .

$\frac{3 x + 1}{x \cdot x} x$

Step 7.2

Cancel the common factor.

$\frac{3 x + 1}{x \cdot x} x$

Step 7.3

Rewrite the expression.

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