Basic Math Examples

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

Step 1

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

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

Step 2

Reduce the expression $\frac{a^{2}}{a^{3}}$ by cancelling the common factors.

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Step 2.1

Multiply by $1$ .

$\sqrt[5]{\frac{a^{2} \cdot 1}{a^{3}}} \sqrt[3]{a}$

Step 2.2

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

$\sqrt[5]{\frac{a^{2} \cdot 1}{a^{2} a}} \sqrt[3]{a}$

Step 2.3

Cancel the common factor.

$\sqrt[5]{\frac{a^{2} \cdot 1}{a^{2} a}} \sqrt[3]{a}$

Step 2.4

Rewrite the expression.

$\sqrt[5]{\frac{1}{a}} \sqrt[3]{a}$

Step 3

Rewrite $\sqrt[5]{\frac{1}{a}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{a}}$ .

$\frac{\sqrt[5]{1}}{\sqrt[5]{a}} \sqrt[3]{a}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[5]{a}} \sqrt[3]{a}$

Step 5

Multiply $\frac{1}{\sqrt[5]{a}}$ by $\frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{4}}$ .

$\frac{1}{\sqrt[5]{a}} \cdot \frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{4}} \sqrt[3]{a}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{1}{\sqrt[5]{a}}$ by $\frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{4}}$ .

$\frac{{\sqrt[5]{a}}^{4}}{\sqrt[5]{a} {\sqrt[5]{a}}^{4}} \sqrt[3]{a}$

Step 6.2

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

$\frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{1} {\sqrt[5]{a}}^{4}} \sqrt[3]{a}$

Step 6.3

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

$\frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{1 + 4}} \sqrt[3]{a}$

Step 6.4

Add $1$ and $4$ .

$\frac{{\sqrt[5]{a}}^{4}}{{\sqrt[5]{a}}^{5}} \sqrt[3]{a}$

Step 6.5

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

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Step 6.5.1

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

$\frac{{\sqrt[5]{a}}^{4}}{{(a^{\frac{1}{5}})}^{5}} \sqrt[3]{a}$

Step 6.5.2

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

$\frac{{\sqrt[5]{a}}^{4}}{a^{\frac{1}{5} \cdot 5}} \sqrt[3]{a}$

Step 6.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{{\sqrt[5]{a}}^{4}}{a^{\frac{5}{5}}} \sqrt[3]{a}$

Step 6.5.4

Cancel the common factor of $5$ .

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Step 6.5.4.1

Cancel the common factor.

$\frac{{\sqrt[5]{a}}^{4}}{a^{\frac{5}{5}}} \sqrt[3]{a}$

Step 6.5.4.2

Rewrite the expression.

$\frac{{\sqrt[5]{a}}^{4}}{a^{1}} \sqrt[3]{a}$

Step 6.5.5

Simplify.

$\frac{{\sqrt[5]{a}}^{4}}{a} \sqrt[3]{a}$

Step 7

Rewrite ${\sqrt[5]{a}}^{4}$ as $\sqrt[5]{a^{4}}$ .

$\frac{\sqrt[5]{a^{4}}}{a} \sqrt[3]{a}$

Step 8

Multiply $\frac{\sqrt[5]{a^{4}}}{a} \sqrt[3]{a}$ .

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Step 8.1

Combine $\frac{\sqrt[5]{a^{4}}}{a}$ and $\sqrt[3]{a}$ .

$\frac{\sqrt[5]{a^{4}} \sqrt[3]{a}}{a}$

Step 8.2

Rewrite the expression using the least common index of $15$ .

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Step 8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{a^{4}}$ as $a^{\frac{4}{5}}$ .

$\frac{a^{\frac{4}{5}} \sqrt[3]{a}}{a}$

Step 8.2.2

Rewrite $a^{\frac{4}{5}}$ as $a^{\frac{12}{15}}$ .

$\frac{a^{\frac{12}{15}} \sqrt[3]{a}}{a}$

Step 8.2.3

Rewrite $a^{\frac{12}{15}}$ as $\sqrt[15]{a^{12}}$ .

$\frac{\sqrt[15]{a^{12}} \sqrt[3]{a}}{a}$

Step 8.2.4

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

$\frac{\sqrt[15]{a^{12}} a^{\frac{1}{3}}}{a}$

Step 8.2.5

Rewrite $a^{\frac{1}{3}}$ as $a^{\frac{5}{15}}$ .

$\frac{\sqrt[15]{a^{12}} a^{\frac{5}{15}}}{a}$

Step 8.2.6

Rewrite $a^{\frac{5}{15}}$ as $\sqrt[15]{a^{5}}$ .

$\frac{\sqrt[15]{a^{12}} \sqrt[15]{a^{5}}}{a}$

Step 8.3

Combine using the product rule for radicals.

$\frac{\sqrt[15]{a^{12} a^{5}}}{a}$

Step 8.4

Multiply $a^{12}$ by $a^{5}$ by adding the exponents.

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Step 8.4.1

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

$\frac{\sqrt[15]{a^{12 + 5}}}{a}$

Step 8.4.2

Add $12$ and $5$ .

$\frac{\sqrt[15]{a^{17}}}{a}$

Step 9

Simplify the numerator.

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Step 9.1

Factor out $a^{15}$ .

$\frac{\sqrt[15]{a^{15} a^{2}}}{a}$

Step 9.2

Pull terms out from under the radical.

$\frac{a \sqrt[15]{a^{2}}}{a}$

Step 10

Cancel the common factor of $a$ .

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Step 10.1

Cancel the common factor.

$\frac{a \sqrt[15]{a^{2}}}{a}$

Step 10.2

Divide $\sqrt[15]{a^{2}}$ by $1$ .

$\sqrt[15]{a^{2}}$