Algebra Examples

$\sqrt[5]{\frac{c}{8 a b^{4}}}$

Step 1

Rewrite $\sqrt[5]{\frac{c}{8 a b^{4}}}$ as $\frac{\sqrt[5]{c}}{\sqrt[5]{8 a b^{4}}}$ .

$\frac{\sqrt[5]{c}}{\sqrt[5]{8 a b^{4}}}$

Step 2

Multiply $\frac{\sqrt[5]{c}}{\sqrt[5]{8 a b^{4}}}$ by $\frac{{\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{4}}$ .

$\frac{\sqrt[5]{c}}{\sqrt[5]{8 a b^{4}}} \cdot \frac{{\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{4}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt[5]{c}}{\sqrt[5]{8 a b^{4}}}$ by $\frac{{\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{4}}$ .

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{\sqrt[5]{8 a b^{4}} {\sqrt[5]{8 a b^{4}}}^{4}}$

Step 3.2

Raise $\sqrt[5]{8 a b^{4}}$ to the power of $1$ .

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{1} {\sqrt[5]{8 a b^{4}}}^{4}}$

Step 3.3

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

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{1 + 4}}$

Step 3.4

Add $1$ and $4$ .

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{\sqrt[5]{8 a b^{4}}}^{5}}$

Step 3.5

Rewrite ${\sqrt[5]{8 a b^{4}}}^{5}$ as $8 a b^{4}$ .

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8 a b^{4}}$ as ${(8 a b^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{({(8 a b^{4})}^{\frac{1}{5}})}^{5}}$

Step 3.5.2

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

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{(8 a b^{4})}^{\frac{1}{5} \cdot 5}}$

Step 3.5.3

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

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{(8 a b^{4})}^{\frac{5}{5}}}$

Step 3.5.4

Cancel the common factor of $5$ .

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Step 3.5.4.1

Cancel the common factor.

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{(8 a b^{4})}^{\frac{5}{5}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{{(8 a b^{4})}^{1}}$

Step 3.5.5

Simplify.

$\frac{\sqrt[5]{c} {\sqrt[5]{8 a b^{4}}}^{4}}{8 a b^{4}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite ${\sqrt[5]{8 a b^{4}}}^{4}$ as $\sqrt[5]{{(8 a b^{4})}^{4}}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{{(8 a b^{4})}^{4}}}{8 a b^{4}}$

Step 4.2

Apply the product rule to $8 a b^{4}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{{(8 a)}^{4} {(b^{4})}^{4}}}{8 a b^{4}}$

Step 4.3

Apply the product rule to $8 a$ .

$\frac{\sqrt[5]{c} \sqrt[5]{8^{4} a^{4} {(b^{4})}^{4}}}{8 a b^{4}}$

Step 4.4

Raise $8$ to the power of $4$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4096 a^{4} {(b^{4})}^{4}}}{8 a b^{4}}$

Step 4.5

Multiply the exponents in ${(b^{4})}^{4}$ .

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Step 4.5.1

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

$\frac{\sqrt[5]{c} \sqrt[5]{4096 a^{4} b^{4 \cdot 4}}}{8 a b^{4}}$

Step 4.5.2

Multiply $4$ by $4$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4096 a^{4} b^{16}}}{8 a b^{4}}$

Step 4.6

Rewrite $4096 a^{4} b^{16}$ as ${(4 b^{3})}^{5} \cdot (4 a^{4} b)$ .

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Step 4.6.1

Factor $1024$ out of $4096$ .

$\frac{\sqrt[5]{c} \sqrt[5]{1024 (4) a^{4} b^{16}}}{8 a b^{4}}$

Step 4.6.2

Rewrite $1024$ as $4^{5}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4^{5} \cdot 4 a^{4} b^{16}}}{8 a b^{4}}$

Step 4.6.3

Factor out $b^{15}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4^{5} \cdot 4 a^{4} (b^{15} b)}}{8 a b^{4}}$

Step 4.6.4

Rewrite $b^{15}$ as ${(b^{3})}^{5}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4^{5} \cdot 4 a^{4} ({(b^{3})}^{5} b)}}{8 a b^{4}}$

Step 4.6.5

Move $a^{4}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4^{5} \cdot 4 {(b^{3})}^{5} a^{4} b}}{8 a b^{4}}$

Step 4.6.6

Move $4$ .

$\frac{\sqrt[5]{c} \sqrt[5]{4^{5} {(b^{3})}^{5} \cdot 4 a^{4} b}}{8 a b^{4}}$

Step 4.6.7

Rewrite $4^{5} {(b^{3})}^{5}$ as ${(4 b^{3})}^{5}$ .

$\frac{\sqrt[5]{c} \sqrt[5]{{(4 b^{3})}^{5} \cdot 4 a^{4} b}}{8 a b^{4}}$

Step 4.6.8

Add parentheses.

$\frac{\sqrt[5]{c} \sqrt[5]{{(4 b^{3})}^{5} \cdot 4 (a^{4} b)}}{8 a b^{4}}$

Step 4.6.9

Add parentheses.

$\frac{\sqrt[5]{c} \sqrt[5]{{(4 b^{3})}^{5} \cdot (4 a^{4} b)}}{8 a b^{4}}$

Step 4.7

Pull terms out from under the radical.

$\frac{\sqrt[5]{c} \cdot 4 b^{3} \sqrt[5]{4 a^{4} b}}{8 a b^{4}}$

Step 4.8

Combine using the product rule for radicals.

$\frac{4 b^{3} \sqrt[5]{c \cdot 4 a^{4} b}}{8 a b^{4}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $4$ and $8$ .

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Step 5.1.1

Factor $4$ out of $4 b^{3} \sqrt[5]{c \cdot 4 a^{4} b}$ .

$\frac{4 (b^{3} \sqrt[5]{c \cdot 4 a^{4} b})}{8 a b^{4}}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $4$ out of $8 a b^{4}$ .

$\frac{4 (b^{3} \sqrt[5]{c \cdot 4 a^{4} b})}{4 (2 a b^{4})}$

Step 5.1.2.2

Cancel the common factor.

$\frac{4 (b^{3} \sqrt[5]{c \cdot 4 a^{4} b})}{4 (2 a b^{4})}$

Step 5.1.2.3

Rewrite the expression.

$\frac{b^{3} \sqrt[5]{c \cdot 4 a^{4} b}}{2 a b^{4}}$

Step 5.2

Cancel the common factor of $b^{3}$ and $b^{4}$ .

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Step 5.2.1

Factor $b^{3}$ out of $b^{3} \sqrt[5]{c \cdot 4 a^{4} b}$ .

$\frac{b^{3} (\sqrt[5]{c \cdot 4 a^{4} b})}{2 a b^{4}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $b^{3}$ out of $2 a b^{4}$ .

$\frac{b^{3} (\sqrt[5]{c \cdot 4 a^{4} b})}{b^{3} (2 a b)}$

Step 5.2.2.2

Cancel the common factor.

$\frac{b^{3} \sqrt[5]{c \cdot 4 a^{4} b}}{b^{3} (2 a b)}$

Step 5.2.2.3

Rewrite the expression.

$\frac{\sqrt[5]{c \cdot 4 a^{4} b}}{2 a b}$

Step 5.3

Reorder factors in $\frac{\sqrt[5]{c \cdot 4 a^{4} b}}{2 a b}$ .

$\frac{\sqrt[5]{4 c a^{4} b}}{2 a b}$