Simplify cube root of x^2* fifth root of x

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

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

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

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

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{10}{15}}$ .

$x^{\frac{10}{15}} \sqrt[5]{x}$

Rewrite $x^{\frac{10}{15}}$ as $\sqrt[15]{x^{10}}$ .

$\sqrt[15]{x^{10}} \sqrt[5]{x}$

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

$\sqrt[15]{x^{10}} x^{\frac{1}{5}}$

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

$\sqrt[15]{x^{10}} x^{\frac{3}{15}}$

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

$\sqrt[15]{x^{10}} \sqrt[15]{x^{3}}$

Combine using the product rule for radicals.

$\sqrt[15]{x^{10} x^{3}}$

Multiply $x^{10}$ by $x^{3}$ by adding the exponents.

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

$\sqrt[15]{x^{10 + 3}}$

Add $10$ and $3$ .

$\sqrt[15]{x^{13}}$