Calculus Examples

$\frac{lim_{x \to - 8} \sqrt[5]{x} - \sqrt[9]{x}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 8$ .

$\frac{lim_{x \to - 8} \sqrt[5]{x} - lim_{x \to - 8} \sqrt[9]{x}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt[5]{lim_{x \to - 8} x} - lim_{x \to - 8} \sqrt[9]{x}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 3

Move the limit under the radical sign.

$\frac{\sqrt[5]{lim_{x \to - 8} x} - \sqrt[9]{lim_{x \to - 8} x}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 4

Evaluate the limits by plugging in $- 8$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt[5]{- 8} - \sqrt[9]{lim_{x \to - 8} x}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 4.2

Evaluate the limit of $x$ by plugging in $- 8$ for $x$ .

$\frac{\sqrt[5]{- 8} - \sqrt[9]{- 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite $- 8$ as ${(- 1)}^{5} \cdot 8$ .

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Step 5.1.1.1

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{\sqrt[5]{- 1 (8)} - \sqrt[9]{- 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.1.2

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$\frac{\sqrt[5]{{(- 1)}^{5} \cdot 8} - \sqrt[9]{- 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.2

Pull terms out from under the radical.

$\frac{- 1 \sqrt[5]{8} - \sqrt[9]{- 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.3

Rewrite $- 1 \sqrt[5]{8}$ as $- \sqrt[5]{8}$ .

$\frac{- \sqrt[5]{8} - \sqrt[9]{- 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.4

Rewrite $- 8$ as ${(- 1)}^{9} \cdot 8$ .

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Step 5.1.4.1

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- \sqrt[5]{8} - \sqrt[9]{- 1 (8)}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.4.2

Rewrite $- 1$ as ${(- 1)}^{9}$ .

$\frac{- \sqrt[5]{8} - \sqrt[9]{{(- 1)}^{9} \cdot 8}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.5

Pull terms out from under the radical.

$\frac{- \sqrt[5]{8} - (- 1 \sqrt[9]{8})}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.6

Rewrite $8$ as $2^{3}$ .

$\frac{- \sqrt[5]{8} - (- 1 \sqrt[9]{2^{3}})}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.7

Rewrite $\sqrt[9]{2^{3}}$ as $\sqrt[3]{\sqrt[3]{2^{3}}}$ .

$\frac{- \sqrt[5]{8} - (- 1 \sqrt[3]{\sqrt[3]{2^{3}}})}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.8

Pull terms out from under the radical, assuming real numbers.

$\frac{- \sqrt[5]{8} - (- 1 \sqrt[3]{2})}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.9

Rewrite $- 1 \sqrt[3]{2}$ as $- \sqrt[3]{2}$ .

$\frac{- \sqrt[5]{8} - - \sqrt[3]{2}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.10

Multiply $- - \sqrt[3]{2}$ .

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Step 5.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt[5]{8} + 1 \sqrt[3]{2}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.1.10.2

Multiply $\sqrt[3]{2}$ by $1$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{\sqrt[5]{x} + \sqrt[9]{x}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{x^{\frac{1}{5}} + \sqrt[9]{x}}$

Step 5.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{x}$ as $x^{\frac{1}{9}}$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{x^{\frac{1}{5}} + x^{\frac{1}{9}}}$

Step 5.2.3

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{1}{5}} + x^{\frac{1}{9}}$ .

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Step 5.2.3.1

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{1}{5}}$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{x^{\frac{1}{9}} x^{\frac{4}{45}} + x^{\frac{1}{9}}}$

Step 5.2.3.2

Multiply by $1$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{x^{\frac{1}{9}} x^{\frac{4}{45}} + x^{\frac{1}{9}} \cdot 1}$

Step 5.2.3.3

Factor $x^{\frac{1}{9}}$ out of $x^{\frac{1}{9}} x^{\frac{4}{45}} + x^{\frac{1}{9}} \cdot 1$ .

$\frac{- \sqrt[5]{8} + \sqrt[3]{2}}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$

Step 5.3

Factor $- 1$ out of $- \sqrt[5]{8}$ .

$\frac{- (\sqrt[5]{8}) + \sqrt[3]{2}}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$

Step 5.4

Factor $- 1$ out of $\sqrt[3]{2}$ .

$\frac{- (\sqrt[5]{8}) - 1 (- \sqrt[3]{2})}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$

Step 5.5

Factor $- 1$ out of $- (\sqrt[5]{8}) - 1 (- \sqrt[3]{2})$ .

$\frac{- (\sqrt[5]{8} - \sqrt[3]{2})}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$

Step 5.6

Rewrite $- (\sqrt[5]{8} - \sqrt[3]{2})$ as $- 1 (\sqrt[5]{8} - \sqrt[3]{2})$ .

$\frac{- 1 (\sqrt[5]{8} - \sqrt[3]{2})}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$

Step 5.7

Move the negative in front of the fraction.

$- \frac{\sqrt[5]{8} - \sqrt[3]{2}}{x^{\frac{1}{9}} (x^{\frac{4}{45}} + 1)}$