Algebra Examples

$\sqrt[10]{r^{- 8} \cdot r^{3}}$

Step 1

Multiply $r^{- 8}$ by $r^{3}$ by adding the exponents.

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Step 1.1

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

$\sqrt[10]{r^{- 8 + 3}}$

Step 1.2

Add $- 8$ and $3$ .

$\sqrt[10]{r^{- 5}}$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[10]{\frac{1}{r^{5}}}$

Step 3

Rewrite $1$ as $1^{5}$ .

$\sqrt[10]{\frac{1^{5}}{r^{5}}}$

Step 4

Rewrite $\frac{1^{5}}{r^{5}}$ as ${(\frac{1}{r})}^{5}$ .

$\sqrt[10]{{(\frac{1}{r})}^{5}}$

Step 5

Rewrite $\sqrt[10]{{(\frac{1}{r})}^{5}}$ as $\sqrt{\sqrt[5]{{(\frac{1}{r})}^{5}}}$ .

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

Step 6

Pull terms out from under the radical.

$\sqrt{\frac{1}{r}}$

Step 7

Rewrite $\sqrt{\frac{1}{r}}$ as $\frac{\sqrt{1}}{\sqrt{r}}$ .

$\frac{\sqrt{1}}{\sqrt{r}}$

Step 8

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{r}}$

Step 9

Multiply $\frac{1}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$\frac{1}{\sqrt{r}} \cdot \frac{\sqrt{r}}{\sqrt{r}}$

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{\sqrt{r}}$ by $\frac{\sqrt{r}}{\sqrt{r}}$ .

$\frac{\sqrt{r}}{\sqrt{r} \sqrt{r}}$

Step 10.2

Raise $\sqrt{r}$ to the power of $1$ .

$\frac{\sqrt{r}}{{\sqrt{r}}^{1} \sqrt{r}}$

Step 10.3

Raise $\sqrt{r}$ to the power of $1$ .

$\frac{\sqrt{r}}{{\sqrt{r}}^{1} {\sqrt{r}}^{1}}$

Step 10.4

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

$\frac{\sqrt{r}}{{\sqrt{r}}^{1 + 1}}$

Step 10.5

Add $1$ and $1$ .

$\frac{\sqrt{r}}{{\sqrt{r}}^{2}}$

Step 10.6

Rewrite ${\sqrt{r}}^{2}$ as $r$ .

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Step 10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r}$ as $r^{\frac{1}{2}}$ .

$\frac{\sqrt{r}}{{(r^{\frac{1}{2}})}^{2}}$

Step 10.6.2

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

$\frac{\sqrt{r}}{r^{\frac{1}{2} \cdot 2}}$

Step 10.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{r}}{r^{\frac{2}{2}}}$

Step 10.6.4

Cancel the common factor of $2$ .

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Step 10.6.4.1

Cancel the common factor.

$\frac{\sqrt{r}}{r^{\frac{2}{2}}}$

Step 10.6.4.2

Rewrite the expression.

$\frac{\sqrt{r}}{r^{1}}$

Step 10.6.5

Simplify.

$\frac{\sqrt{r}}{r}$