Calculus Examples

$lim_{x \to - 8} \sqrt[3]{x} - \sqrt[7]{x}$

Step 1

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

$lim_{x \to - 8} \sqrt[3]{x} - lim_{x \to - 8} \sqrt[7]{x}$

Step 2

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to - 8} x} - lim_{x \to - 8} \sqrt[7]{x}$

Step 3

Move the limit under the radical sign.

$\sqrt[3]{lim_{x \to - 8} x} - \sqrt[7]{lim_{x \to - 8} x}$

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 $x$ .

$\sqrt[3]{- 8} - \sqrt[7]{lim_{x \to - 8} x}$

Step 4.2

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

$\sqrt[3]{- 8} - \sqrt[7]{- 8}$

Step 5

Simplify each term.

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Step 5.1

Rewrite $- 8$ as ${(- 2)}^{3}$ .

$\sqrt[3]{{(- 2)}^{3}} - \sqrt[7]{- 8}$

Step 5.2

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

$- 2 - \sqrt[7]{- 8}$

Step 5.3

Rewrite $- 8$ as ${(- 1)}^{7} \cdot 8$ .

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Step 5.3.1

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

$- 2 - \sqrt[7]{- 1 (8)}$

Step 5.3.2

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$- 2 - \sqrt[7]{{(- 1)}^{7} \cdot 8}$

Step 5.4

Pull terms out from under the radical.

$- 2 - (- 1 \sqrt[7]{8})$

Step 5.5

Rewrite $- 1 \sqrt[7]{8}$ as $- \sqrt[7]{8}$ .

$- 2 - - \sqrt[7]{8}$

Step 5.6

Multiply $- - \sqrt[7]{8}$ .

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Step 5.6.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt[7]{8}$

Step 5.6.2

Multiply $\sqrt[7]{8}$ by $1$ .

$- 2 + \sqrt[7]{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt[7]{8}$

Decimal Form:

$- 0.65409980 \dots$