Simplify the Radical Expression 1/(( square root of 2)^5)

$\frac{1}{{(\sqrt{2})}^{5}}$

Simplify the denominator.

Rewrite ${\sqrt{2}}^{5}$ as ${(2^{5})}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{2^{5}}}$

Raise $2$ to the power of $5$ .

$\frac{1}{\sqrt{32}}$

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

Factor $16$ out of $32$ .

$\frac{1}{\sqrt{16 (2)}}$

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

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

Pull terms out from under the radical.

$\frac{1}{| 4 | \sqrt{2}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

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

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{4 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{4 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Move $\sqrt{2}$ .

$\frac{\sqrt{2}}{4 (\sqrt{2} \sqrt{2})}$

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

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

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

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

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

$\frac{\sqrt{2}}{4 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{2}}{4 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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

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

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

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

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2}}{4 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $4$ by $2$ .

$\frac{\sqrt{2}}{8}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{8}$

Decimal Form:

$0.17677669 \dots$

Simplify the Radical Expression 1/(( square root of 2)^5)

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