Simplify the Radical Expression ( fourth root of 32)^2

${(\sqrt[4]{32})}^{2}$

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

Factor $16$ out of $32$ .

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Apply the product rule to $2 \sqrt[4]{2}$ .

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

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

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

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

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

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

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

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

Combine $\frac{1}{4}$ and $2$ .

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$4 \sqrt{2}$

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{2}$

Decimal Form:

$5.65685424 \dots$