Trigonometry Examples

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

Step 1

Rewrite $\sqrt[4]{\frac{1}{8}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{8}}$ .

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

Step 2

Any root of $1$ is $1$ .

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

Step 3

Multiply $\frac{1}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{1}{\sqrt[4]{8}} \cdot \frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt[4]{8}}$ by $\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{3}}$ .

$\frac{{\sqrt[4]{8}}^{3}}{\sqrt[4]{8} {\sqrt[4]{8}}^{3}}$

Step 4.2

Raise $\sqrt[4]{8}$ to the power of $1$ .

$\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1} {\sqrt[4]{8}}^{3}}$

Step 4.3

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

$\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{1 + 3}}$

Step 4.4

Add $1$ and $3$ .

$\frac{{\sqrt[4]{8}}^{3}}{{\sqrt[4]{8}}^{4}}$

Step 4.5

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

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{8}}^{3}}{{(8^{\frac{1}{4}})}^{4}}$

Step 4.5.2

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

$\frac{{\sqrt[4]{8}}^{3}}{8^{\frac{1}{4} \cdot 4}}$

Step 4.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{{\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Step 4.5.4

Cancel the common factor of $4$ .

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Step 4.5.4.1

Cancel the common factor.

$\frac{{\sqrt[4]{8}}^{3}}{8^{\frac{4}{4}}}$

Step 4.5.4.2

Rewrite the expression.

$\frac{{\sqrt[4]{8}}^{3}}{8^{1}}$

Step 4.5.5

Evaluate the exponent.

$\frac{{\sqrt[4]{8}}^{3}}{8}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[4]{8}}^{3}$ as $\sqrt[4]{8^{3}}$ .

$\frac{\sqrt[4]{8^{3}}}{8}$

Step 5.2

Raise $8$ to the power of $3$ .

$\frac{\sqrt[4]{512}}{8}$

Step 5.3

Rewrite $512$ as $4^{4} \cdot 2$ .

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Step 5.3.1

Factor $256$ out of $512$ .

$\frac{\sqrt[4]{256 (2)}}{8}$

Step 5.3.2

Rewrite $256$ as $4^{4}$ .

$\frac{\sqrt[4]{4^{4} \cdot 2}}{8}$

Step 5.4

Pull terms out from under the radical.

$\frac{4 \sqrt[4]{2}}{8}$

Step 6

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

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Step 6.1

Factor $4$ out of $4 \sqrt[4]{2}$ .

$\frac{4 (\sqrt[4]{2})}{8}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $4$ out of $8$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.59460355 \dots$