Simplify the Radical Expression 20 root of 1/(b*b^7)

$\sqrt[20]{\frac{1}{b \cdot b^{7}}}$

Step 1

Multiply $b$ by $b^{7}$ by adding the exponents.

Multiply $b$ by $b^{7}$ .

Raise $b$ to the power of $1$ .

$\sqrt[20]{\frac{1}{b^{1} \cdot b^{7}}}$

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

$\sqrt[20]{\frac{1}{b^{1 + 7}}}$

Add $1$ and $7$ .

$\sqrt[20]{\frac{1}{b^{8}}}$

Step 2

Rewrite $1$ as $1^{4}$ .

$\sqrt[20]{\frac{1^{4}}{b^{8}}}$

Step 3

Rewrite $b^{8}$ as ${(b^{2})}^{4}$ .

$\sqrt[20]{\frac{1^{4}}{{(b^{2})}^{4}}}$

Step 4

Rewrite $\frac{1^{4}}{{(b^{2})}^{4}}$ as ${(\frac{1}{b^{2}})}^{4}$ .

$\sqrt[20]{{(\frac{1}{b^{2}})}^{4}}$

Step 5

Rewrite $\sqrt[20]{{(\frac{1}{b^{2}})}^{4}}$ as $\sqrt[5]{\sqrt[4]{{(\frac{1}{b^{2}})}^{4}}}$ .

$\sqrt[5]{\sqrt[4]{{(\frac{1}{b^{2}})}^{4}}}$

Step 6

Pull terms out from under the radical.

$\sqrt[5]{| \frac{1}{b^{2}} |}$

Step 7

Remove non-negative terms from the absolute value.

$\sqrt[5]{\frac{| 1 |}{b^{2}}}$

Step 8

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

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

Step 9

Rewrite $\sqrt[5]{\frac{1}{b^{2}}}$ as $\frac{\sqrt[5]{1}}{\sqrt[5]{b^{2}}}$ .

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

Step 10

Any root of $1$ is $1$ .

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

Step 11

Multiply $\frac{1}{\sqrt[5]{b^{2}}}$ by $\frac{{\sqrt[5]{b^{2}}}^{4}}{{\sqrt[5]{b^{2}}}^{4}}$ .

$\frac{1}{\sqrt[5]{b^{2}}} \cdot \frac{{\sqrt[5]{b^{2}}}^{4}}{{\sqrt[5]{b^{2}}}^{4}}$

Step 12

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[5]{b^{2}}}$ by $\frac{{\sqrt[5]{b^{2}}}^{4}}{{\sqrt[5]{b^{2}}}^{4}}$ .

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

Raise $\sqrt[5]{b^{2}}$ to the power of $1$ .

$\frac{{\sqrt[5]{b^{2}}}^{4}}{{\sqrt[5]{b^{2}}}^{1} {\sqrt[5]{b^{2}}}^{4}}$

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

$\frac{{\sqrt[5]{b^{2}}}^{4}}{{\sqrt[5]{b^{2}}}^{1 + 4}}$

Add $1$ and $4$ .

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

Rewrite ${\sqrt[5]{b^{2}}}^{5}$ as $b^{2}$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{b^{2}}$ as $b^{\frac{2}{5}}$ .

$\frac{{\sqrt[5]{b^{2}}}^{4}}{{(b^{\frac{2}{5}})}^{5}}$

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

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

Combine $\frac{2}{5}$ and $5$ .

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

Multiply $2$ by $5$ .

$\frac{{\sqrt[5]{b^{2}}}^{4}}{b^{\frac{10}{5}}}$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

$\frac{{\sqrt[5]{b^{2}}}^{4}}{b^{\frac{5 \cdot 2}{5 (1)}}}$

Cancel the common factor.

$\frac{{\sqrt[5]{b^{2}}}^{4}}{b^{\frac{5 \cdot 2}{5 \cdot 1}}}$

Rewrite the expression.

$\frac{{\sqrt[5]{b^{2}}}^{4}}{b^{\frac{2}{1}}}$

Divide $2$ by $1$ .

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

Step 13

Simplify the numerator.

Rewrite ${\sqrt[5]{b^{2}}}^{4}$ as ${({(b^{2})}^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[5]{{(b^{2})}^{4}}}{b^{2}}$

Multiply the exponents in ${(b^{2})}^{4}$ .

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

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

Multiply $2$ by $4$ .

$\frac{\sqrt[5]{b^{8}}}{b^{2}}$

Factor out $b^{5}$ .

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

Pull terms out from under the radical.

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

Step 14

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the Radical Expression 20 root of 1/(b*b^7)

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