Algebra Examples

$\frac{\sqrt[5]{r^{6}}}{\sqrt[3]{r^{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor out $r^{5}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Combine and simplify the denominator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Add $1$ and $2$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

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

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

Step 3.1.5.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 3.1.5.4

Multiply $2$ by $3$ .

$\frac{r \sqrt[5]{r} {\sqrt[3]{r^{2}}}^{2}}{r^{\frac{6}{3}}}$

Step 3.1.5.5

Cancel the common factor of $6$ and $3$ .

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Step 3.1.5.5.1

Factor $3$ out of $6$ .

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

Step 3.1.5.5.2

Cancel the common factors.

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Step 3.1.5.5.2.1

Factor $3$ out of $3$ .

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

Step 3.1.5.5.2.2

Cancel the common factor.

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

Step 3.1.5.5.2.3

Rewrite the expression.

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

Step 3.1.5.5.2.4

Divide $2$ by $1$ .

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

Step 3.2

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

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Step 3.2.1

Factor $r$ out of $r \sqrt[5]{r} {\sqrt[3]{r^{2}}}^{2}$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $r$ out of $r^{2}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $2$ .

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

Step 4.3

Factor out $r^{3}$ .

$\frac{\sqrt[5]{r} \sqrt[3]{r^{3} r}}{r}$

Step 4.4

Pull terms out from under the radical.

$\frac{\sqrt[5]{r} r \sqrt[3]{r}}{r}$

Step 4.5

Combine exponents.

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Step 4.5.1

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

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Step 4.5.1.1

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

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

Step 4.5.1.2

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

$\frac{r (r^{\frac{3}{15}} \sqrt[3]{r})}{r}$

Step 4.5.1.3

Rewrite $r^{\frac{3}{15}}$ as $\sqrt[15]{r^{3}}$ .

$\frac{r (\sqrt[15]{r^{3}} \sqrt[3]{r})}{r}$

Step 4.5.1.4

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

$\frac{r (\sqrt[15]{r^{3}} r^{\frac{1}{3}})}{r}$

Step 4.5.1.5

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

$\frac{r (\sqrt[15]{r^{3}} r^{\frac{5}{15}})}{r}$

Step 4.5.1.6

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

$\frac{r (\sqrt[15]{r^{3}} \sqrt[15]{r^{5}})}{r}$

Step 4.5.2

Combine using the product rule for radicals.

$\frac{r \sqrt[15]{r^{3} r^{5}}}{r}$

Step 4.5.3

Multiply $r^{3}$ by $r^{5}$ by adding the exponents.

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Step 4.5.3.1

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

$\frac{r \sqrt[15]{r^{3 + 5}}}{r}$

Step 4.5.3.2

Add $3$ and $5$ .

$\frac{r \sqrt[15]{r^{8}}}{r}$

Step 5

Cancel the common factor of $r$ .

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Step 5.1

Cancel the common factor.

$\frac{r \sqrt[15]{r^{8}}}{r}$

Step 5.2

Divide $\sqrt[15]{r^{8}}$ by $1$ .

$\sqrt[15]{r^{8}}$