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

$\sqrt[4]{2} \cdot \sqrt[6]{2^{5}}$

Step 1

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

$\sqrt[4]{2} \cdot \sqrt[6]{32}$

Step 2

Multiply $\sqrt[4]{2} \sqrt[6]{32}$ .

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

$2^{\frac{1}{4}} \sqrt[6]{32}$

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

$2^{\frac{3}{12}} \sqrt[6]{32}$

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

$\sqrt[12]{2^{3}} \sqrt[6]{32}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{32}$ as $32^{\frac{1}{6}}$ .

$\sqrt[12]{2^{3}} \cdot 32^{\frac{1}{6}}$

Rewrite $32^{\frac{1}{6}}$ as $32^{\frac{2}{12}}$ .

$\sqrt[12]{2^{3}} \cdot 32^{\frac{2}{12}}$

Rewrite $32^{\frac{2}{12}}$ as $\sqrt[12]{32^{2}}$ .

$\sqrt[12]{2^{3}} \sqrt[12]{32^{2}}$

Combine using the product rule for radicals.

$\sqrt[12]{2^{3} \cdot 32^{2}}$

Rewrite $32$ as $2^{5}$ .

$\sqrt[12]{2^{3} \cdot {(2^{5})}^{2}}$

Multiply the exponents in ${(2^{5})}^{2}$ .

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

$\sqrt[12]{2^{3} \cdot 2^{5 \cdot 2}}$

Multiply $5$ by $2$ .

$\sqrt[12]{2^{3} \cdot 2^{10}}$

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

$\sqrt[12]{2^{3 + 10}}$

Add $3$ and $10$ .

$\sqrt[12]{2^{13}}$

Step 3

Raise $2$ to the power of $13$ .

$\sqrt[12]{8192}$

Step 4

Rewrite $8192$ as $2^{12} \cdot 2$ .

Factor $4096$ out of $8192$ .

$\sqrt[12]{4096 (2)}$

Rewrite $4096$ as $2^{12}$ .

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

Step 5

Pull terms out from under the radical.

$2 \sqrt[12]{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[12]{2}$

Decimal Form:

$2.11892618 \dots$

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

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