Simplify square root of 2^3* fifth root of 2^4

$\sqrt{2^{3}} \cdot \sqrt[5]{2^{4}}$

Raise $2$ to the power of $3$ .

$\sqrt{8} \cdot \sqrt[5]{2^{4}}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

$2 \sqrt{2} \cdot \sqrt[5]{16}$

Multiply $2 \sqrt{2} \sqrt[5]{16}$ .

Rewrite the expression using the least common index of $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{16}$ as $16^{\frac{1}{5}}$ .

$2 (16^{\frac{1}{5}} \sqrt{2})$

Rewrite $16^{\frac{1}{5}}$ as $16^{\frac{2}{10}}$ .

$2 (16^{\frac{2}{10}} \sqrt{2})$

Rewrite $16^{\frac{2}{10}}$ as $\sqrt[10]{16^{2}}$ .

$2 (\sqrt[10]{16^{2}} \sqrt{2})$

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

$2 (\sqrt[10]{16^{2}} \cdot 2^{\frac{1}{2}})$

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{5}{10}}$ .

$2 (\sqrt[10]{16^{2}} \cdot 2^{\frac{5}{10}})$

Rewrite $2^{\frac{5}{10}}$ as $\sqrt[10]{2^{5}}$ .

$2 (\sqrt[10]{16^{2}} \sqrt[10]{2^{5}})$

Combine using the product rule for radicals.

$2 \sqrt[10]{16^{2} \cdot 2^{5}}$

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

$2 \sqrt[10]{{(2^{4})}^{2} \cdot 2^{5}}$

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

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

$2 \sqrt[10]{2^{4 \cdot 2} \cdot 2^{5}}$

Multiply $4$ by $2$ .

$2 \sqrt[10]{2^{8} \cdot 2^{5}}$

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

$2 \sqrt[10]{2^{8 + 5}}$

Add $8$ and $5$ .

$2 \sqrt[10]{2^{13}}$

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

$2 \sqrt[10]{8192}$

Rewrite $8192$ as $2^{10} \cdot 8$ .

Factor $1024$ out of $8192$ .

$2 \sqrt[10]{1024 (8)}$

Rewrite $1024$ as $2^{10}$ .

$2 \sqrt[10]{2^{10} \cdot 8}$

Pull terms out from under the radical.

$2 (2 \sqrt[10]{8})$

Multiply $2$ by $2$ .

$4 \sqrt[10]{8}$

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt[10]{8}$

Decimal Form:

$4.92457765 \dots$

Simplify square root of 2^3* fifth root of 2^4

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