Simplify ( square root of 2^5)/( cube root of 2)

$\frac{\sqrt{2^{5}}}{\sqrt[3]{2}}$

Step 1

Simplify the numerator.

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

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

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

Factor $16$ out of $32$ .

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

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

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

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

Factor $2$ out of $4 \sqrt{2} {\sqrt[3]{2}}^{2}$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

Rewrite the expression using the least common index of $6$ .

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

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

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$2 (4^{\frac{2}{6}} \sqrt{2})$

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$2 (\sqrt[6]{4^{2}} \sqrt{2})$

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

$2 (\sqrt[6]{4^{2}} \cdot 2^{\frac{1}{2}})$

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

$2 (\sqrt[6]{4^{2}} \cdot 2^{\frac{3}{6}})$

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$2 (\sqrt[6]{4^{2}} \sqrt[6]{2^{3}})$

Combine using the product rule for radicals.

$2 \sqrt[6]{4^{2} \cdot 2^{3}}$

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

$2 \sqrt[6]{{(2^{2})}^{2} \cdot 2^{3}}$

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

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

$2 \sqrt[6]{2^{2 \cdot 2} \cdot 2^{3}}$

Multiply $2$ by $2$ .

$2 \sqrt[6]{2^{4} \cdot 2^{3}}$

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

$2 \sqrt[6]{2^{4 + 3}}$

Add $4$ and $3$ .

$2 \sqrt[6]{2^{7}}$

Step 8

Raise $2$ to the power of $7$ .

$2 \sqrt[6]{128}$

Step 9

Rewrite $128$ as $2^{6} \cdot 2$ .

Factor $64$ out of $128$ .

$2 \sqrt[6]{64 (2)}$

Rewrite $64$ as $2^{6}$ .

$2 \sqrt[6]{2^{6} \cdot 2}$

Step 10

Pull terms out from under the radical.

$2 (2 \sqrt[6]{2})$

Step 11

Multiply $2$ by $2$ .

$4 \sqrt[6]{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[6]{2}$

Decimal Form:

$4.48984819 \dots$

Simplify ( square root of 2^5)/( cube root of 2)

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