Simplify cube root of y^7 cube root of 81y^8

$\sqrt[3]{y^{7}} \sqrt[3]{81 y^{8}}$

Step 1

Rewrite $y^{7}$ as ${(y^{2})}^{3} y$ .

Factor out $y^{6}$ .

$\sqrt[3]{y^{6} y} \sqrt[3]{81 y^{8}}$

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

$\sqrt[3]{{(y^{2})}^{3} y} \sqrt[3]{81 y^{8}}$

Step 2

Pull terms out from under the radical.

$y^{2} \sqrt[3]{y} \sqrt[3]{81 y^{8}}$

Step 3

Rewrite $81 y^{8}$ as ${(3 y^{2})}^{3} \cdot (3 y^{2})$ .

Factor $27$ out of $81$ .

$y^{2} \sqrt[3]{y} \sqrt[3]{27 (3) y^{8}}$

Rewrite $27$ as $3^{3}$ .

$y^{2} \sqrt[3]{y} \sqrt[3]{3^{3} \cdot 3 y^{8}}$

Factor out $y^{6}$ .

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

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

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

Move $3$ .

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

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

Move $y^{2}$ .

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

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

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

Add $2$ and $2$ .

$3 y^{4} \sqrt[3]{y} \sqrt[3]{3 y^{2}}$

Step 7

Multiply $3 y^{4} \sqrt[3]{y} \sqrt[3]{3 y^{2}}$ .

Combine using the product rule for radicals.

$3 y^{4} \sqrt[3]{3 y^{2} y}$

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

$3 y^{4} \sqrt[3]{3 (y^{1} y^{2})}$

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

$3 y^{4} \sqrt[3]{3 y^{1 + 2}}$

Add $1$ and $2$ .

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

Step 8

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

$3 y^{4} \sqrt[3]{y^{3} \cdot 3}$

Step 9

Pull terms out from under the radical.

$3 y^{4} (y \sqrt[3]{3})$

Step 10

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

Move $y$ .

$3 (y \cdot y^{4}) \sqrt[3]{3}$

Multiply $y$ by $y^{4}$ .

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

$3 (y^{1} y^{4}) \sqrt[3]{3}$

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

$3 y^{1 + 4} \sqrt[3]{3}$

Add $1$ and $4$ .

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

Simplify cube root of y^7 cube root of 81y^8

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