Simplify cube root of y^10 cube root of 81y^11

$\sqrt[3]{y^{10}} \sqrt[3]{81 y^{11}}$

Step 1

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

Factor out $y^{9}$ .

$\sqrt[3]{y^{9} y} \sqrt[3]{81 y^{11}}$

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

$\sqrt[3]{{(y^{3})}^{3} y} \sqrt[3]{81 y^{11}}$

Step 2

Pull terms out from under the radical.

$y^{3} \sqrt[3]{y} \sqrt[3]{81 y^{11}}$

Step 3

Rewrite $81 y^{11}$ as ${(3 y^{3})}^{3} \cdot (3 y^{2})$ .

Factor $27$ out of $81$ .

$y^{3} \sqrt[3]{y} \sqrt[3]{27 (3) y^{11}}$

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

$y^{3} \sqrt[3]{y} \sqrt[3]{3^{3} \cdot 3 y^{11}}$

Factor out $y^{9}$ .

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

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

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

Move $3$ .

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

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

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

Move $y^{3}$ .

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

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

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

Add $3$ and $3$ .

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

Step 7

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

Combine using the product rule for radicals.

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

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

$3 y^{6} \sqrt[3]{3 (y^{1} y^{2})}$

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

$3 y^{6} \sqrt[3]{3 y^{1 + 2}}$

Add $1$ and $2$ .

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

Step 8

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $y^{6}$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y^{6}$ .

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

$3 (y^{1} y^{6}) \sqrt[3]{3}$

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

$3 y^{1 + 6} \sqrt[3]{3}$

Add $1$ and $6$ .

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

Simplify cube root of y^10 cube root of 81y^11

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