Simplify the Radical Expression ( fourth root of 32)/( cube root of 2)

$\frac{\sqrt[4]{32}}{\sqrt[3]{2}}$

Simplify the numerator.

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

Factor $16$ out of $32$ .

$\frac{\sqrt[4]{16 (2)}}{\sqrt[3]{2}}$

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

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

Pull terms out from under the radical.

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

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

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

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

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

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

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

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

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

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

Multiply $\sqrt[4]{2} \sqrt[3]{4}$ .

Rewrite the expression using the least common index of $12$ .

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

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

Rewrite $2^{\frac{1}{4}}$ as $2^{\frac{3}{12}}$ .

$2^{\frac{3}{12}} \sqrt[3]{4}$

Rewrite $2^{\frac{3}{12}}$ as $\sqrt[12]{2^{3}}$ .

$\sqrt[12]{2^{3}} \sqrt[3]{4}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\sqrt[12]{2^{3}} \cdot 4^{\frac{1}{3}}$

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{4}{12}}$ .

$\sqrt[12]{2^{3}} \cdot 4^{\frac{4}{12}}$

Rewrite $4^{\frac{4}{12}}$ as $\sqrt[12]{4^{4}}$ .

$\sqrt[12]{2^{3}} \sqrt[12]{4^{4}}$

Combine using the product rule for radicals.

$\sqrt[12]{2^{3} \cdot 4^{4}}$

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

$\sqrt[12]{2^{3} \cdot {(2^{2})}^{4}}$

Multiply the exponents in ${(2^{2})}^{4}$ .

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

$\sqrt[12]{2^{3} \cdot 2^{2 \cdot 4}}$

Multiply $2$ by $4$ .

$\sqrt[12]{2^{3} \cdot 2^{8}}$

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

$\sqrt[12]{2^{3 + 8}}$

Add $3$ and $8$ .

$\sqrt[12]{2^{11}}$

Raise $2$ to the power of $11$ .

$\sqrt[12]{2048}$

The result can be shown in multiple forms.

Exact Form:

$\sqrt[12]{2048}$

Decimal Form:

$1.88774862 \dots$

Simplify the Radical Expression ( fourth root of 32)/( cube root of 2)

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