Finite Math Examples

$\sqrt[3]{\frac{81 x^{5} (y^{3})}{\sqrt[3]{3 x^{2}}}}$

Step 1

Multiply $\frac{81 x^{5} y^{3}}{\sqrt[3]{3 x^{2}}}$ by $\frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}$ .

$\sqrt[3]{\frac{81 x^{5} y^{3}}{\sqrt[3]{3 x^{2}}} \cdot \frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{81 x^{5} y^{3}}{\sqrt[3]{3 x^{2}}}$ by $\frac{{\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{2}}$ .

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{\sqrt[3]{3 x^{2}} {\sqrt[3]{3 x^{2}}}^{2}}}$

Step 2.2

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

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{1} {\sqrt[3]{3 x^{2}}}^{2}}}$

Step 2.3

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

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{1 + 2}}}$

Step 2.4

Add $1$ and $2$ .

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{\sqrt[3]{3 x^{2}}}^{3}}}$

Step 2.5

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

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x^{2}}$ as ${(3 x^{2})}^{\frac{1}{3}}$ .

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{({(3 x^{2})}^{\frac{1}{3}})}^{3}}}$

Step 2.5.2

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

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{1}{3} \cdot 3}}}$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{3}{3}}}}$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{\frac{3}{3}}}}$

Step 2.5.4.2

Rewrite the expression.

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{{(3 x^{2})}^{1}}}$

Step 2.5.5

Simplify.

$\sqrt[3]{\frac{81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{3 x^{2}}}$

Step 3

Cancel the common factor of $81$ and $3$ .

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Step 3.1

Factor $3$ out of $81 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}$ .

$\sqrt[3]{\frac{3 (27 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2})}{3 x^{2}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $3 x^{2}$ .

$\sqrt[3]{\frac{3 (27 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2})}{3 (x^{2})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt[3]{\frac{3 (27 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2})}{3 x^{2}}}$

Step 3.2.3

Rewrite the expression.

$\sqrt[3]{\frac{27 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{x^{2}}}$

Step 4

Cancel the common factor of $x^{5}$ and $x^{2}$ .

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Step 4.1

Factor $x^{2}$ out of $27 x^{5} y^{3} {\sqrt[3]{3 x^{2}}}^{2}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\sqrt[3]{\frac{x^{2} (27 x^{3} y^{3} {\sqrt[3]{3 x^{2}}}^{2})}{x^{2} \cdot 1}}$

Step 4.2.2

Cancel the common factor.

$\sqrt[3]{\frac{x^{2} (27 x^{3} y^{3} {\sqrt[3]{3 x^{2}}}^{2})}{x^{2} \cdot 1}}$

Step 4.2.3

Rewrite the expression.

$\sqrt[3]{\frac{27 x^{3} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}{1}}$

Step 4.2.4

Divide $27 x^{3} y^{3} {\sqrt[3]{3 x^{2}}}^{2}$ by $1$ .

$\sqrt[3]{27 x^{3} y^{3} {\sqrt[3]{3 x^{2}}}^{2}}$

Step 5

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

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

Step 6

Apply the product rule to $3 x^{2}$ .

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

Step 7

Raise $3$ to the power of $2$ .

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{9 {(x^{2})}^{2}}}$

Step 8

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

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Step 8.1

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

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{9 x^{2 \cdot 2}}}$

Step 8.2

Multiply $2$ by $2$ .

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{9 x^{4}}}$

Step 9

Rewrite $9 x^{4}$ as $x^{3} \cdot (9 x)$ .

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Step 9.1

Factor out $x^{3}$ .

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{9 (x^{3} x)}}$

Step 9.2

Reorder $9$ and $x^{3}$ .

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{x^{3} \cdot 9 x}}$

Step 9.3

Add parentheses.

$\sqrt[3]{27 x^{3} y^{3} \sqrt[3]{x^{3} \cdot (9 x)}}$

Step 10

Pull terms out from under the radical.

$\sqrt[3]{27 x^{3} y^{3} (x \sqrt[3]{9 x})}$

Step 11

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

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Step 11.1

Move $x$ .

$\sqrt[3]{27 (x \cdot x^{3}) y^{3} \sqrt[3]{9 x}}$

Step 11.2

Multiply $x$ by $x^{3}$ .

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Step 11.2.1

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

$\sqrt[3]{27 (x^{1} x^{3}) y^{3} \sqrt[3]{9 x}}$

Step 11.2.2

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

$\sqrt[3]{27 x^{1 + 3} y^{3} \sqrt[3]{9 x}}$

Step 11.3

Add $1$ and $3$ .

$\sqrt[3]{27 x^{4} y^{3} \sqrt[3]{9 x}}$

Step 12

Rewrite $27 x^{4} y^{3} \sqrt[3]{9 x}$ as ${(3 x y)}^{3} (x \sqrt[3]{9 x})$ .

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Step 12.1

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

$\sqrt[3]{3^{3} x^{4} y^{3} \sqrt[3]{9 x}}$

Step 12.2

Factor out $x^{3}$ .

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

Step 12.3

Move $x$ .

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

Step 12.4

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

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

Step 12.5

Add parentheses.

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

Step 13

Pull terms out from under the radical.

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