Simplify the Radical Expression sixth root of y^5 cube root of y^5

$\sqrt[6]{y^{5}} \sqrt[3]{y^{5}}$

Step 1

Factor out $y^{3}$ .

$\sqrt[6]{y^{5}} \sqrt[3]{y^{3} y^{2}}$

Step 2

Pull terms out from under the radical.

$\sqrt[6]{y^{5}} (y \sqrt[3]{y^{2}})$

Step 3

Multiply $\sqrt[6]{y^{5}} (y \sqrt[3]{y^{2}})$ .

Rewrite the expression using the least common index of $6$ .

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

$y (\sqrt[6]{y^{5}} y^{\frac{2}{3}})$

Rewrite $y^{\frac{2}{3}}$ as $y^{\frac{4}{6}}$ .

$y (\sqrt[6]{y^{5}} y^{\frac{4}{6}})$

Rewrite $y^{\frac{4}{6}}$ as $\sqrt[6]{y^{4}}$ .

$y (\sqrt[6]{y^{5}} \sqrt[6]{y^{4}})$

Combine using the product rule for radicals.

$y \sqrt[6]{y^{5} y^{4}}$

Multiply $y^{5}$ by $y^{4}$ by adding the exponents.

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

$y \sqrt[6]{y^{5 + 4}}$

Add $5$ and $4$ .

$y \sqrt[6]{y^{9}}$

Step 4

Factor out $y^{6}$ .

$y \sqrt[6]{y^{6} y^{3}}$

Step 5

Pull terms out from under the radical.

$y (| y | \sqrt[6]{y^{3}})$

Step 6

Rewrite $\sqrt[6]{y^{3}}$ as $\sqrt{\sqrt[3]{y^{3}}}$ .

$y (| y | \sqrt{\sqrt[3]{y^{3}}})$

Step 7

Pull terms out from under the radical.

$y | y | \sqrt{y}$