Algebra Examples

$\sqrt[3]{y^{13}} \sqrt[3]{81 y^{14}}$

Step 1

Rewrite $y^{13}$ as ${(y^{4})}^{3} y$ .

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Step 1.1

Factor out $y^{12}$ .

$\sqrt[3]{y^{12} y} \sqrt[3]{81 y^{14}}$

Step 1.2

Rewrite $y^{12}$ as ${(y^{4})}^{3}$ .

$\sqrt[3]{{(y^{4})}^{3} y} \sqrt[3]{81 y^{14}}$

Step 2

Pull terms out from under the radical.

$y^{4} \sqrt[3]{y} \sqrt[3]{81 y^{14}}$

Step 3

Rewrite $81 y^{14}$ as ${(3 y^{4})}^{3} \cdot (3 y^{2})$ .

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Step 3.1

Factor $27$ out of $81$ .

$y^{4} \sqrt[3]{y} \sqrt[3]{27 (3) y^{14}}$

Step 3.2

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

$y^{4} \sqrt[3]{y} \sqrt[3]{3^{3} \cdot 3 y^{14}}$

Step 3.3

Factor out $y^{12}$ .

$y^{4} \sqrt[3]{y} \sqrt[3]{3^{3} \cdot 3 (y^{12} y^{2})}$

Step 3.4

Rewrite $y^{12}$ as ${(y^{4})}^{3}$ .

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

Step 3.5

Move $3$ .

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

Step 3.6

Rewrite $3^{3} {(y^{4})}^{3}$ as ${(3 y^{4})}^{3}$ .

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

Step 3.7

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

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Step 6.1

Move $y^{4}$ .

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

Step 6.2

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

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

Step 6.3

Add $4$ and $4$ .

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

Step 7

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

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

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

$3 y^{8} \sqrt[3]{3 (y^{1} y^{2})}$

Step 7.3

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

$3 y^{8} \sqrt[3]{3 y^{1 + 2}}$

Step 7.4

Add $1$ and $2$ .

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

Step 8

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $y^{8}$ by $y$ by adding the exponents.

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Step 10.1

Move $y$ .

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

Step 10.2

Multiply $y$ by $y^{8}$ .

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Step 10.2.1

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

$3 (y^{1} y^{8}) \sqrt[3]{3}$

Step 10.2.2

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

$3 y^{1 + 8} \sqrt[3]{3}$

Step 10.3

Add $1$ and $8$ .

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