Algebra Examples

$\sqrt[7]{{(3 c)}^{5}}$

Step 1

Apply the product rule to $3 c$ .

$\sqrt[7]{3^{5} c^{5}}$

Step 2

Raise $3$ to the power of $5$ .

$\sqrt[7]{243 c^{5}}$

Step 3

Rewrite $\sqrt[7]{243 c^{5}}$ as $\sqrt[7]{243} \sqrt[7]{c^{5}}$ .

$\sqrt[7]{243} \sqrt[7]{c^{5}}$

Step 4

Multiply to rationalize the numerator.

$\frac{\sqrt[7]{c^{5}} (\sqrt[7]{243} {\sqrt[7]{243}}^{6})}{{\sqrt[7]{243}}^{6}}$

Step 5

Simplify.

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Step 5.1

Raise $\sqrt[7]{243}$ to the power of $1$ .

$\frac{\sqrt[7]{c^{5}} ({\sqrt[7]{243}}^{1} {\sqrt[7]{243}}^{6})}{{\sqrt[7]{243}}^{6}}$

Step 5.2

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

$\frac{\sqrt[7]{c^{5}} {\sqrt[7]{243}}^{1 + 6}}{{\sqrt[7]{243}}^{6}}$

Step 5.3

Add $1$ and $6$ .

$\frac{\sqrt[7]{c^{5}} {\sqrt[7]{243}}^{7}}{{\sqrt[7]{243}}^{6}}$

Step 5.4

Rewrite ${\sqrt[7]{243}}^{7}$ as $243$ .

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Step 5.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{243}$ as $243^{\frac{1}{7}}$ .

$\frac{\sqrt[7]{c^{5}} {(243^{\frac{1}{7}})}^{7}}{{\sqrt[7]{243}}^{6}}$

Step 5.4.2

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

$\frac{\sqrt[7]{c^{5}} \cdot 243^{\frac{1}{7} \cdot 7}}{{\sqrt[7]{243}}^{6}}$

Step 5.4.3

Combine $\frac{1}{7}$ and $7$ .

$\frac{\sqrt[7]{c^{5}} \cdot 243^{\frac{7}{7}}}{{\sqrt[7]{243}}^{6}}$

Step 5.4.4

Cancel the common factor of $7$ .

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Step 5.4.4.1

Cancel the common factor.

$\frac{\sqrt[7]{c^{5}} \cdot 243^{\frac{7}{7}}}{{\sqrt[7]{243}}^{6}}$

Step 5.4.4.2

Rewrite the expression.

$\frac{\sqrt[7]{c^{5}} \cdot 243^{1}}{{\sqrt[7]{243}}^{6}}$

Step 5.4.5

Evaluate the exponent.

$\frac{\sqrt[7]{c^{5}} \cdot 243}{{\sqrt[7]{243}}^{6}}$

Step 6

Multiply to rationalize the numerator.

$\frac{243 (\sqrt[7]{c^{5}} {\sqrt[7]{c^{5}}}^{6})}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7

Simplify.

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Step 7.1

Raise $\sqrt[7]{c^{5}}$ to the power of $1$ .

$\frac{243 ({\sqrt[7]{c^{5}}}^{1} {\sqrt[7]{c^{5}}}^{6})}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.2

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

$\frac{243 {\sqrt[7]{c^{5}}}^{1 + 6}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.3

Add $1$ and $6$ .

$\frac{243 {\sqrt[7]{c^{5}}}^{7}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4

Rewrite ${\sqrt[7]{c^{5}}}^{7}$ as $c^{5}$ .

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Step 7.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{c^{5}}$ as $c^{\frac{5}{7}}$ .

$\frac{243 {(c^{\frac{5}{7}})}^{7}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.2

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

$\frac{243 c^{\frac{5}{7} \cdot 7}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.3

Combine $\frac{5}{7}$ and $7$ .

$\frac{243 c^{\frac{5 \cdot 7}{7}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.4

Multiply $5$ by $7$ .

$\frac{243 c^{\frac{35}{7}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.5

Cancel the common factor of $35$ and $7$ .

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Step 7.4.5.1

Factor $7$ out of $35$ .

$\frac{243 c^{\frac{7 \cdot 5}{7}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.5.2

Cancel the common factors.

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Step 7.4.5.2.1

Factor $7$ out of $7$ .

$\frac{243 c^{\frac{7 \cdot 5}{7 (1)}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.5.2.2

Cancel the common factor.

$\frac{243 c^{\frac{7 \cdot 5}{7 \cdot 1}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.5.2.3

Rewrite the expression.

$\frac{243 c^{\frac{5}{1}}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.4.5.2.4

Divide $5$ by $1$ .

$\frac{243 c^{5}}{{\sqrt[7]{243}}^{6} {\sqrt[7]{c^{5}}}^{6}}$

Step 7.5

Start simplifying.

$\frac{243 c^{5}}{{\sqrt[7]{243}}^{6} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.6

Rewrite ${\sqrt[7]{243}}^{6}$ as $\sqrt[7]{243^{6}}$ .

$\frac{243 c^{5}}{\sqrt[7]{243^{6}} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.7

Raise $243$ to the power of $6$ .

$\frac{243 c^{5}}{\sqrt[7]{205891132094649} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.8

Rewrite $205891132094649$ as $81^{7} \cdot 9$ .

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Step 7.8.1

Factor $22876792454961$ out of $205891132094649$ .

$\frac{243 c^{5}}{\sqrt[7]{22876792454961 (9)} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.8.2

Rewrite $22876792454961$ as $81^{7}$ .

$\frac{243 c^{5}}{\sqrt[7]{81^{7} \cdot 9} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.9

Pull terms out from under the radical.

$\frac{243 c^{5}}{81 \sqrt[7]{9} \sqrt[7]{{(c^{5})}^{6}}}$

Step 7.10

Multiply the exponents in ${(c^{5})}^{6}$ .

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Step 7.10.1

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

$\frac{243 c^{5}}{81 \sqrt[7]{9} \sqrt[7]{c^{5 \cdot 6}}}$

Step 7.10.2

Multiply $5$ by $6$ .

$\frac{243 c^{5}}{81 \sqrt[7]{9} \sqrt[7]{c^{30}}}$

Step 7.11

Rewrite $c^{30}$ as ${(c^{4})}^{7} c^{2}$ .

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Step 7.11.1

Factor out $c^{28}$ .

$\frac{243 c^{5}}{81 \sqrt[7]{9} \sqrt[7]{c^{28} c^{2}}}$

Step 7.11.2

Rewrite $c^{28}$ as ${(c^{4})}^{7}$ .

$\frac{243 c^{5}}{81 \sqrt[7]{9} \sqrt[7]{{(c^{4})}^{7} c^{2}}}$

Step 7.12

Pull terms out from under the radical.

$\frac{243 c^{5}}{81 \sqrt[7]{9} (c^{4} \sqrt[7]{c^{2}})}$

Step 7.13

Combine using the product rule for radicals.

$\frac{243 c^{5}}{81 c^{4} \sqrt[7]{9 c^{2}}}$