Simplify sixth root of 2* square root of 2^5

$\sqrt[6]{2} \cdot \sqrt{2^{5}}$

Step 1

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

$\sqrt[6]{2} \cdot \sqrt{32}$

Step 2

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

Factor $16$ out of $32$ .

$\sqrt[6]{2} \cdot \sqrt{16 (2)}$

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $2^{3}$ by adding the exponents.

Multiply $2$ by $2^{3}$ .

Raise $2$ to the power of $1$ .

$4 \sqrt[6]{2^{1} \cdot 2^{3}}$

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

$4 \sqrt[6]{2^{1 + 3}}$

Add $1$ and $3$ .

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

Step 5

Raise $2$ to the power of $4$ .

$4 \sqrt[6]{16}$

Step 6

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

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

Step 7

Rewrite $\sqrt[6]{4^{2}}$ as $\sqrt[3]{\sqrt{4^{2}}}$ .

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

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$4 \sqrt[3]{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[3]{4}$

Decimal Form:

$6.34960420 \dots$

Simplify sixth root of 2* square root of 2^5

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