Simplify eighth root of 1/(w^3*w^-1)

$\sqrt[8]{\frac{1}{w^{3} \cdot w^{- 1}}}$

Step 1

Multiply $w^{3}$ by $w^{- 1}$ by adding the exponents.

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

$\sqrt[8]{\frac{1}{w^{3 - 1}}}$

Subtract $1$ from $3$ .

$\sqrt[8]{\frac{1}{w^{2}}}$

Step 2

Rewrite $1$ as $1^{2}$ .

$\sqrt[8]{\frac{1^{2}}{w^{2}}}$

Step 3

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

$\sqrt[8]{{(\frac{1}{w})}^{2}}$

Step 4

Rewrite $\sqrt[8]{{(\frac{1}{w})}^{2}}$ as $\sqrt[4]{\sqrt{{(\frac{1}{w})}^{2}}}$ .

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

Step 5

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

$\sqrt[4]{\frac{1}{w}}$

Step 6

Rewrite $\sqrt[4]{\frac{1}{w}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{w}}$ .

$\frac{\sqrt[4]{1}}{\sqrt[4]{w}}$

Step 7

Any root of $1$ is $1$ .

$\frac{1}{\sqrt[4]{w}}$

Step 8

Multiply $\frac{1}{\sqrt[4]{w}}$ by $\frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{3}}$ .

$\frac{1}{\sqrt[4]{w}} \cdot \frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{3}}$

Step 9

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt[4]{w}}$ by $\frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{3}}$ .

$\frac{{\sqrt[4]{w}}^{3}}{\sqrt[4]{w} {\sqrt[4]{w}}^{3}}$

Raise $\sqrt[4]{w}$ to the power of $1$ .

$\frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{1} {\sqrt[4]{w}}^{3}}$

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

$\frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{{\sqrt[4]{w}}^{3}}{{\sqrt[4]{w}}^{4}}$

Rewrite ${\sqrt[4]{w}}^{4}$ as $w$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{w}$ as $w^{\frac{1}{4}}$ .

$\frac{{\sqrt[4]{w}}^{3}}{{(w^{\frac{1}{4}})}^{4}}$

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

$\frac{{\sqrt[4]{w}}^{3}}{w^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{{\sqrt[4]{w}}^{3}}{w^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{{\sqrt[4]{w}}^{3}}{w^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{{\sqrt[4]{w}}^{3}}{w^{1}}$

Simplify.

$\frac{{\sqrt[4]{w}}^{3}}{w}$

Step 10

Rewrite ${\sqrt[4]{w}}^{3}$ as ${(w^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[4]{w^{3}}}{w}$

Simplify eighth root of 1/(w^3*w^-1)

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