Simplify cube root of y^4 cube root of 16y^5

$\sqrt[3]{y^{4}} \sqrt[3]{16 y^{5}}$

Step 1

Factor out $y^{3}$ .

$\sqrt[3]{y^{3} y} \sqrt[3]{16 y^{5}}$

Step 2

Pull terms out from under the radical.

$y \sqrt[3]{y} \sqrt[3]{16 y^{5}}$

Step 3

Rewrite $16 y^{5}$ as ${(2 y)}^{3} \cdot (2 y^{2})$ .

Factor $8$ out of $16$ .

$y \sqrt[3]{y} \sqrt[3]{8 (2) y^{5}}$

Rewrite $8$ as $2^{3}$ .

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

Factor out $y^{3}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 7

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

Combine using the product rule for radicals.

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

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

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

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

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

Add $1$ and $2$ .

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

Step 8

Reorder $2$ and $y^{3}$ .

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

Move $y$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Simplify cube root of y^4 cube root of 16y^5

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