Algebra Examples

$\sqrt[28]{\frac{1}{p^{6} \cdot p}}$

Step 1

Multiply $p^{6}$ by $p$ by adding the exponents.

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Step 1.1

Multiply $p^{6}$ by $p$ .

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Step 1.1.1

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

$\sqrt[28]{\frac{1}{p^{6} \cdot p^{1}}}$

Step 1.1.2

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

$\sqrt[28]{\frac{1}{p^{6 + 1}}}$

Step 1.2

Add $6$ and $1$ .

$\sqrt[28]{\frac{1}{p^{7}}}$

Step 2

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

$\sqrt[28]{\frac{1^{7}}{p^{7}}}$

Step 3

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

$\sqrt[28]{{(\frac{1}{p})}^{7}}$

Step 4

Rewrite $\sqrt[28]{{(\frac{1}{p})}^{7}}$ as $\sqrt[4]{\sqrt[7]{{(\frac{1}{p})}^{7}}}$ .

$\sqrt[4]{\sqrt[7]{{(\frac{1}{p})}^{7}}}$

Step 5

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

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

Step 6

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

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

Step 7

Any root of $1$ is $1$ .

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Add $1$ and $3$ .

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

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

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

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

Step 9.5.3

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

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

Step 9.5.4

Cancel the common factor of $4$ .

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Step 9.5.4.1

Cancel the common factor.

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

Step 9.5.4.2

Rewrite the expression.

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

Step 9.5.5

Simplify.

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

Step 10

Rewrite ${\sqrt[4]{p}}^{3}$ as $\sqrt[4]{p^{3}}$ .

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