Basic Math Examples

$\frac{\sqrt[3]{c^{2} d}}{\sqrt[3]{5 d^{2}}}$

Step 1

Combine $\sqrt[3]{c^{2} d}$ and $\sqrt[3]{5 d^{2}}$ into a single radical.

$\sqrt[3]{\frac{c^{2} d}{5 d^{2}}}$

Step 2

Reduce the expression $\frac{c^{2} d}{5 d^{2}}$ by cancelling the common factors.

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Step 2.1

Factor $d$ out of $c^{2} d$ .

$\sqrt[3]{\frac{d c^{2}}{5 d^{2}}}$

Step 2.2

Factor $d$ out of $5 d^{2}$ .

$\sqrt[3]{\frac{d c^{2}}{d (5 d)}}$

Step 2.3

Cancel the common factor.

$\sqrt[3]{\frac{d c^{2}}{d (5 d)}}$

Step 2.4

Rewrite the expression.

$\sqrt[3]{\frac{c^{2}}{5 d}}$

Step 3

Rewrite $\sqrt[3]{\frac{c^{2}}{5 d}}$ as $\frac{\sqrt[3]{c^{2}}}{\sqrt[3]{5 d}}$ .

$\frac{\sqrt[3]{c^{2}}}{\sqrt[3]{5 d}}$

Step 4

Multiply $\frac{\sqrt[3]{c^{2}}}{\sqrt[3]{5 d}}$ by $\frac{{\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{2}}$ .

$\frac{\sqrt[3]{c^{2}}}{\sqrt[3]{5 d}} \cdot \frac{{\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{\sqrt[3]{c^{2}}}{\sqrt[3]{5 d}}$ by $\frac{{\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{2}}$ .

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{\sqrt[3]{5 d} {\sqrt[3]{5 d}}^{2}}$

Step 5.2

Raise $\sqrt[3]{5 d}$ to the power of $1$ .

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{1} {\sqrt[3]{5 d}}^{2}}$

Step 5.3

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

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{\sqrt[3]{5 d}}^{3}}$

Step 5.5

Rewrite ${\sqrt[3]{5 d}}^{3}$ as $5 d$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 d}$ as ${(5 d)}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{({(5 d)}^{\frac{1}{3}})}^{3}}$

Step 5.5.2

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

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{(5 d)}^{\frac{1}{3} \cdot 3}}$

Step 5.5.3

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

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{(5 d)}^{\frac{3}{3}}}$

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{(5 d)}^{\frac{3}{3}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{{(5 d)}^{1}}$

Step 5.5.5

Simplify.

$\frac{\sqrt[3]{c^{2}} {\sqrt[3]{5 d}}^{2}}{5 d}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[3]{5 d}}^{2}$ as $\sqrt[3]{{(5 d)}^{2}}$ .

$\frac{\sqrt[3]{c^{2}} \sqrt[3]{{(5 d)}^{2}}}{5 d}$

Step 6.2

Apply the product rule to $5 d$ .

$\frac{\sqrt[3]{c^{2}} \sqrt[3]{5^{2} d^{2}}}{5 d}$

Step 6.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[3]{c^{2}} \sqrt[3]{25 d^{2}}}{5 d}$

Step 7

Combine using the product rule for radicals.

$\frac{\sqrt[3]{c^{2} \cdot 25 d^{2}}}{5 d}$

Step 8

Reorder factors in $\frac{\sqrt[3]{c^{2} \cdot 25 d^{2}}}{5 d}$ .

$\frac{\sqrt[3]{25 c^{2} d^{2}}}{5 d}$