Simplify cube root of y^10 ninth root of y^6

$\sqrt[3]{y^{10}} \sqrt[9]{y^{6}}$

Step 1

Rewrite $y^{10}$ as ${(y^{3})}^{3} y$ .

Factor out $y^{9}$ .

$\sqrt[3]{y^{9} y} \sqrt[9]{y^{6}}$

Rewrite $y^{9}$ as ${(y^{3})}^{3}$ .

$\sqrt[3]{{(y^{3})}^{3} y} \sqrt[9]{y^{6}}$

Step 2

Pull terms out from under the radical.

$y^{3} \sqrt[3]{y} \sqrt[9]{y^{6}}$

Step 3

Rewrite $y^{6}$ as ${(y^{2})}^{3}$ .

$y^{3} \sqrt[3]{y} \sqrt[9]{{(y^{2})}^{3}}$

Step 4

Rewrite $\sqrt[9]{{(y^{2})}^{3}}$ as $\sqrt[3]{\sqrt[3]{{(y^{2})}^{3}}}$ .

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

Step 5

Pull terms out from under the radical, assuming real numbers.

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

Step 6

Multiply $y^{3} \sqrt[3]{y} \sqrt[3]{y^{2}}$ .

Combine using the product rule for radicals.

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

Multiply $y^{2}$ by $y$ by adding the exponents.

Multiply $y^{2}$ by $y$ .

Raise $y$ to the power of $1$ .

$y^{3} \sqrt[3]{y^{2} y^{1}}$

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

$y^{3} \sqrt[3]{y^{2 + 1}}$

Add $2$ and $1$ .

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

Step 7

Pull terms out from under the radical, assuming real numbers.

$y^{3} y$

Step 8

Multiply $y^{3}$ by $y$ by adding the exponents.

Multiply $y^{3}$ by $y$ .

Raise $y$ to the power of $1$ .

$y^{3} y^{1}$

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

$y^{3 + 1}$

Add $3$ and $1$ .

$y^{4}$