Simplify fourth root of 1/32

$\sqrt[4]{\frac{1}{32}}$

Rewrite $\sqrt[4]{\frac{1}{32}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{32}}$ .

$\frac{\sqrt[4]{1}}{\sqrt[4]{32}}$

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[4]{32}}$

Simplify the denominator.

Rewrite $32$ as $2^{4} \cdot 2$ .

Factor $16$ out of $32$ .

$\frac{1}{\sqrt[4]{16 (2)}}$

Rewrite $16$ as $2^{4}$ .

$\frac{1}{\sqrt[4]{2^{4} \cdot 2}}$

Pull terms out from under the radical.

$\frac{1}{2 \sqrt[4]{2}}$

Multiply $\frac{1}{2 \sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$\frac{1}{2 \sqrt[4]{2}} \cdot \frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{2 \sqrt[4]{2}}$ by $\frac{{\sqrt[4]{2}}^{3}}{{\sqrt[4]{2}}^{3}}$ .

$\frac{{\sqrt[4]{2}}^{3}}{2 \sqrt[4]{2} {\sqrt[4]{2}}^{3}}$

Move $\sqrt[4]{2}$ .

$\frac{{\sqrt[4]{2}}^{3}}{2 (\sqrt[4]{2} {\sqrt[4]{2}}^{3})}$

Raise $\sqrt[4]{2}$ to the power of $1$ .

$\frac{{\sqrt[4]{2}}^{3}}{2 ({\sqrt[4]{2}}^{1} {\sqrt[4]{2}}^{3})}$

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

$\frac{{\sqrt[4]{2}}^{3}}{2 {\sqrt[4]{2}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{{\sqrt[4]{2}}^{3}}{2 {\sqrt[4]{2}}^{4}}$

Rewrite ${\sqrt[4]{2}}^{4}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2}$ as $2^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{2}}^{3}}{2 {(2^{\frac{1}{4}})}^{4}}$

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

$\frac{{\sqrt[4]{2}}^{3}}{2 \cdot 2^{\frac{1}{4} \cdot 4}}$

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

$\frac{{\sqrt[4]{2}}^{3}}{2 \cdot 2^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{{\sqrt[4]{2}}^{3}}{2 \cdot 2^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{{\sqrt[4]{2}}^{3}}{2 \cdot 2^{1}}$

Evaluate the exponent.

$\frac{{\sqrt[4]{2}}^{3}}{2 \cdot 2}$

Simplify the numerator.

Rewrite ${\sqrt[4]{2}}^{3}$ as ${(2^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{2^{3}}}{2 \cdot 2}$

Raise $2$ to the power of $3$ .

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

Multiply $2$ by $2$ .

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.42044820 \dots$

Simplify fourth root of 1/32

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