Evaluate cube root of 2* fifth root of 2

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

Step 1

Multiply $\sqrt[3]{2} \sqrt[5]{2}$ .

Rewrite the expression using the least common index of $15$ .

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

$2^{\frac{1}{3}} \sqrt[5]{2}$

Rewrite $2^{\frac{1}{3}}$ as $2^{\frac{5}{15}}$ .

$2^{\frac{5}{15}} \sqrt[5]{2}$

Rewrite $2^{\frac{5}{15}}$ as $\sqrt[15]{2^{5}}$ .

$\sqrt[15]{2^{5}} \sqrt[5]{2}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2}$ as $2^{\frac{1}{5}}$ .

$\sqrt[15]{2^{5}} \cdot 2^{\frac{1}{5}}$

Rewrite $2^{\frac{1}{5}}$ as $2^{\frac{3}{15}}$ .

$\sqrt[15]{2^{5}} \cdot 2^{\frac{3}{15}}$

Rewrite $2^{\frac{3}{15}}$ as $\sqrt[15]{2^{3}}$ .

$\sqrt[15]{2^{5}} \sqrt[15]{2^{3}}$

Combine using the product rule for radicals.

$\sqrt[15]{2^{5} \cdot 2^{3}}$

Multiply $2^{5}$ by $2^{3}$ by adding the exponents.

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

$\sqrt[15]{2^{5 + 3}}$

Add $5$ and $3$ .

$\sqrt[15]{2^{8}}$

Step 2

Raise $2$ to the power of $8$ .

$\sqrt[15]{256}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt[15]{256}$

Decimal Form:

$1.44726923 \dots$