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

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

Step 1

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

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

Step 2

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

Factor $16$ out of $32$ .

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

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

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

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

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

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

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

Rewrite $2^{\frac{4}{12}}$ as $\sqrt[12]{2^{4}}$ .

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2^{4}$ by $2^{3}$ by adding the exponents.

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

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

Add $4$ and $3$ .

$2 \sqrt[12]{2^{7}}$

Step 5

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

$2 \sqrt[12]{128}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[12]{128}$

Decimal Form:

$2.99661415 \dots$