Finite Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $81 x$ as $3^{3} \cdot (3 x)$ .

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Step 1.1.1

Factor $27$ out of $81$ .

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

Step 1.1.2

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

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

Step 1.1.3

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $3$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 5

Move $3$ to the left of $y^{2}$ .

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