Finite Math Examples

$\sqrt[3]{x y^{5}} \cdot \sqrt[3]{x^{7} y^{20}}$

Step 1

Rewrite $x y^{5}$ as $y^{3} (x y^{2})$ .

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Step 1.1

Factor out $y^{3}$ .

$\sqrt[3]{x (y^{3} y^{2})} \cdot \sqrt[3]{x^{7} y^{20}}$

Step 1.2

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

$\sqrt[3]{y^{3} x y^{2}} \cdot \sqrt[3]{x^{7} y^{20}}$

Step 1.3

Add parentheses.

$\sqrt[3]{y^{3} (x y^{2})} \cdot \sqrt[3]{x^{7} y^{20}}$

Step 2

Pull terms out from under the radical.

$y \sqrt[3]{x y^{2}} \cdot \sqrt[3]{x^{7} y^{20}}$

Step 3

Rewrite $x^{7} y^{20}$ as ${(x^{2} y^{6})}^{3} (x y^{2})$ .

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Step 3.1

Factor out $x^{6}$ .

$y \sqrt[3]{x y^{2}} \cdot \sqrt[3]{x^{6} x y^{20}}$

Step 3.2

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

$y \sqrt[3]{x y^{2}} \cdot \sqrt[3]{{(x^{2})}^{3} x y^{20}}$

Step 3.3

Factor out $y^{18}$ .

$y \sqrt[3]{x y^{2}} \cdot \sqrt[3]{{(x^{2})}^{3} x (y^{18} y^{2})}$

Step 3.4

Rewrite $y^{18}$ as ${(y^{6})}^{3}$ .

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

Step 3.5

Move $x$ .

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

Step 3.6

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

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

Step 3.7

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $y^{6}$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

Add $6$ and $1$ .

$y^{7} \sqrt[3]{x y^{2}} \cdot (x^{2} \sqrt[3]{x y^{2}})$

Step 6

Multiply $y^{7} \sqrt[3]{x y^{2}} (x^{2} \sqrt[3]{x y^{2}})$ .

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Step 6.1

Raise $\sqrt[3]{x y^{2}}$ to the power of $1$ .

$y^{7} (x^{2} ({\sqrt[3]{x y^{2}}}^{1} \sqrt[3]{x y^{2}}))$

Step 6.2

Raise $\sqrt[3]{x y^{2}}$ to the power of $1$ .

$y^{7} (x^{2} ({\sqrt[3]{x y^{2}}}^{1} {\sqrt[3]{x y^{2}}}^{1}))$

Step 6.3

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

$y^{7} (x^{2} {\sqrt[3]{x y^{2}}}^{1 + 1})$

Step 6.4

Add $1$ and $1$ .

$y^{7} (x^{2} {\sqrt[3]{x y^{2}}}^{2})$

Step 7

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

$y^{7} (x^{2} \sqrt[3]{{(x y^{2})}^{2}})$

Step 8

Apply the product rule to $x y^{2}$ .

$y^{7} (x^{2} \sqrt[3]{x^{2} {(y^{2})}^{2}})$

Step 9

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y^{7} (x^{2} \sqrt[3]{x^{2} y^{2 \cdot 2}})$

Step 9.2

Multiply $2$ by $2$ .

$y^{7} (x^{2} \sqrt[3]{x^{2} y^{4}})$

Step 10

Rewrite $x^{2} y^{4}$ as $y^{3} (x^{2} y)$ .

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Step 10.1

Factor out $y^{3}$ .

$y^{7} (x^{2} \sqrt[3]{x^{2} (y^{3} y)})$

Step 10.2

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

$y^{7} (x^{2} \sqrt[3]{y^{3} x^{2} y})$

Step 10.3

Add parentheses.

$y^{7} (x^{2} \sqrt[3]{y^{3} (x^{2} y)})$

Step 11

Pull terms out from under the radical.

$y^{7} (x^{2} (y \sqrt[3]{x^{2} y}))$

Step 12

Multiply $y^{7}$ by $y$ by adding the exponents.

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Step 12.1

Move $y$ .

$y \cdot y^{7} (x^{2} (\sqrt[3]{x^{2} y}))$

Step 12.2

Multiply $y$ by $y^{7}$ .

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Step 12.2.1

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

$y^{1} y^{7} (x^{2} (\sqrt[3]{x^{2} y}))$

Step 12.2.2

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

$y^{1 + 7} (x^{2} (\sqrt[3]{x^{2} y}))$

Step 12.3

Add $1$ and $7$ .

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

Step 13

Reorder factors in $y^{8} (x^{2} (\sqrt[3]{x^{2} y}))$ .

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