Calculus Examples

$lim_{n \to 8} \frac{1}{\sqrt[5]{n}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Limits Quotient Rule on the limit as $n$ approaches $8$ .

$\frac{lim_{n \to 8} 1}{lim_{n \to 8} \sqrt[5]{n}}$

Step 1.2

Evaluate the limit of $1$ which is constant as $n$ approaches $8$ .

$\frac{1}{lim_{n \to 8} \sqrt[5]{n}}$

Step 1.3

Move the limit under the radical sign.

$\frac{1}{\sqrt[5]{lim_{n \to 8} n}}$

Step 2

Evaluate the limit of $n$ by plugging in $8$ for $n$ .

$\frac{1}{\sqrt[5]{8}}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $\frac{1}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$\frac{1}{\sqrt[5]{8}} \cdot \frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$

Step 3.2

Combine and simplify the denominator.

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Step 3.2.1

Multiply $\frac{1}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$\frac{{\sqrt[5]{8}}^{4}}{\sqrt[5]{8} {\sqrt[5]{8}}^{4}}$

Step 3.2.2

Raise $\sqrt[5]{8}$ to the power of $1$ .

$\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1} {\sqrt[5]{8}}^{4}}$

Step 3.2.3

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

$\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1 + 4}}$

Step 3.2.4

Add $1$ and $4$ .

$\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{5}}$

Step 3.2.5

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

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Step 3.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$\frac{{\sqrt[5]{8}}^{4}}{{(8^{\frac{1}{5}})}^{5}}$

Step 3.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{\sqrt[5]{8}}^{4}}{8^{\frac{1}{5} \cdot 5}}$

Step 3.2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{{\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 3.2.5.4

Cancel the common factor of $5$ .

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Step 3.2.5.4.1

Cancel the common factor.

$\frac{{\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 3.2.5.4.2

Rewrite the expression.

$\frac{{\sqrt[5]{8}}^{4}}{8^{1}}$

Step 3.2.5.5

Evaluate the exponent.

$\frac{{\sqrt[5]{8}}^{4}}{8}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Rewrite ${\sqrt[5]{8}}^{4}$ as $\sqrt[5]{8^{4}}$ .

$\frac{\sqrt[5]{8^{4}}}{8}$

Step 3.3.2

Raise $8$ to the power of $4$ .

$\frac{\sqrt[5]{4096}}{8}$

Step 3.3.3

Rewrite $4096$ as $4^{5} \cdot 4$ .

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Step 3.3.3.1

Factor $1024$ out of $4096$ .

$\frac{\sqrt[5]{1024 (4)}}{8}$

Step 3.3.3.2

Rewrite $1024$ as $4^{5}$ .

$\frac{\sqrt[5]{4^{5} \cdot 4}}{8}$

Step 3.3.4

Pull terms out from under the radical.

$\frac{4 \sqrt[5]{4}}{8}$

Step 3.4

Cancel the common factor of $4$ and $8$ .

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Step 3.4.1

Factor $4$ out of $4 \sqrt[5]{4}$ .

$\frac{4 (\sqrt[5]{4})}{8}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $4$ out of $8$ .

$\frac{4 \sqrt[5]{4}}{4 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{4 \sqrt[5]{4}}{4 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{\sqrt[5]{4}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[5]{4}}{2}$

Decimal Form:

$0.65975395 \dots$