Algebra Examples

$\sqrt[21]{c^{- 3} \cdot c^{- 4}}$

Step 1

Multiply $c^{- 3}$ by $c^{- 4}$ 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.

$\sqrt[21]{c^{- 3 - 4}}$

Step 1.2

Subtract $4$ from $- 3$ .

$\sqrt[21]{c^{- 7}}$

Step 2

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

$\sqrt[21]{\frac{1}{c^{7}}}$

Step 3

Rewrite $1$ as $1^{7}$ .

$\sqrt[21]{\frac{1^{7}}{c^{7}}}$

Step 4

Rewrite $\frac{1^{7}}{c^{7}}$ as ${(\frac{1}{c})}^{7}$ .

$\sqrt[21]{{(\frac{1}{c})}^{7}}$

Step 5

Rewrite $\sqrt[21]{{(\frac{1}{c})}^{7}}$ as $\sqrt[3]{\sqrt[7]{{(\frac{1}{c})}^{7}}}$ .

$\sqrt[3]{\sqrt[7]{{(\frac{1}{c})}^{7}}}$

Step 6

Pull terms out from under the radical, assuming real numbers.

$\sqrt[3]{\frac{1}{c}}$

Step 7

Rewrite $\sqrt[3]{\frac{1}{c}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{c}}$ .

$\frac{\sqrt[3]{1}}{\sqrt[3]{c}}$

Step 8

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[3]{c}}$

Step 9

Multiply $\frac{1}{\sqrt[3]{c}}$ by $\frac{{\sqrt[3]{c}}^{2}}{{\sqrt[3]{c}}^{2}}$ .

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

Step 10

Combine and simplify the denominator.

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Step 10.1

Multiply $\frac{1}{\sqrt[3]{c}}$ by $\frac{{\sqrt[3]{c}}^{2}}{{\sqrt[3]{c}}^{2}}$ .

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $2$ .

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

Step 10.5

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

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Step 10.5.1

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

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

Step 10.5.2

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

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

Step 10.5.3

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

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

Step 10.5.4

Cancel the common factor of $3$ .

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Step 10.5.4.1

Cancel the common factor.

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

Step 10.5.4.2

Rewrite the expression.

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

Step 10.5.5

Simplify.

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

Step 11

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

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