Basic Math Examples

$\frac{\sqrt[3]{5 y^{4}}}{\sqrt[3]{15 c^{4}}}$

Step 1

Combine $\sqrt[3]{5 y^{4}}$ and $\sqrt[3]{15 c^{4}}$ into a single radical.

$\sqrt[3]{\frac{5 y^{4}}{15 c^{4}}}$

Step 2

Reduce the expression $\frac{5 y^{4}}{15 c^{4}}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $5 y^{4}$ .

$\sqrt[3]{\frac{5 (y^{4})}{15 c^{4}}}$

Step 2.2

Factor $5$ out of $15 c^{4}$ .

$\sqrt[3]{\frac{5 (y^{4})}{5 (3 c^{4})}}$

Step 2.3

Cancel the common factor.

$\sqrt[3]{\frac{5 y^{4}}{5 (3 c^{4})}}$

Step 2.4

Rewrite the expression.

$\sqrt[3]{\frac{y^{4}}{3 c^{4}}}$

Step 3

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

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Step 3.1

Factor the perfect power $y^{3}$ out of $y^{4}$ .

$\sqrt[3]{\frac{y^{3} y}{3 c^{4}}}$

Step 3.2

Factor the perfect power $c^{3}$ out of $3 c^{4}$ .

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

Step 3.3

Rearrange the fraction $\frac{y^{3} y}{c^{3} \cdot (3 c)}$ .

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

$\frac{y}{c} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{3 c}}$

Step 6

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

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

Step 7

Combine and simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $2$ .

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

Step 7.5

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

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.5.3

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

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

Step 7.5.4

Cancel the common factor of $3$ .

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Step 7.5.4.1

Cancel the common factor.

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

Step 7.5.4.2

Rewrite the expression.

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

Step 7.5.5

Simplify.

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

Step 8

Combine.

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

Step 9

Multiply $c$ by $c$ by adding the exponents.

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Step 9.1

Move $c$ .

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

Step 9.2

Multiply $c$ by $c$ .

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

Step 10

Simplify the numerator.

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Step 10.1

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

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

Step 10.2

Apply the product rule to $3 c$ .

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

Step 10.3

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

$\frac{y \sqrt[3]{y} \sqrt[3]{9 c^{2}}}{c^{2} \cdot 3}$

Step 10.4

Combine using the product rule for radicals.

$\frac{y \sqrt[3]{9 c^{2} y}}{c^{2} \cdot 3}$

Step 11

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

$\frac{y \sqrt[3]{9 c^{2} y}}{3 c^{2}}$