Simplify cube root of y* fifth root of y^3

$\sqrt[3]{y} \cdot \sqrt[5]{y^{3}}$

Step 1

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

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

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

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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{3}}$ as $y^{\frac{3}{5}}$ .

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

Rewrite $y^{\frac{3}{5}}$ as $y^{\frac{9}{15}}$ .

$\sqrt[15]{y^{5}} y^{\frac{9}{15}}$

Rewrite $y^{\frac{9}{15}}$ as $\sqrt[15]{y^{9}}$ .

$\sqrt[15]{y^{5}} \sqrt[15]{y^{9}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[15]{y^{5} y^{9}}$

Step 3

Multiply $y^{5}$ by $y^{9}$ by adding the exponents.

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

$\sqrt[15]{y^{5 + 9}}$

Add $5$ and $9$ .

$\sqrt[15]{y^{14}}$