Algebra Examples

$\frac{6 p}{\sqrt{4 p^{7}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 p^{7}$ as ${(2 p^{3})}^{2} p$ .

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Step 1.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{6 p}{\sqrt{2^{2} p^{7}}}$

Step 1.1.2

Factor out $p^{6}$ .

$\frac{6 p}{\sqrt{2^{2} (p^{6} p)}}$

Step 1.1.3

Rewrite $p^{6}$ as ${(p^{3})}^{2}$ .

$\frac{6 p}{\sqrt{2^{2} ({(p^{3})}^{2} p)}}$

Step 1.1.4

Rewrite $2^{2} {(p^{3})}^{2}$ as ${(2 p^{3})}^{2}$ .

$\frac{6 p}{\sqrt{{(2 p^{3})}^{2} p}}$

Step 1.2

Pull terms out from under the radical.

$\frac{6 p}{2 p^{3} \sqrt{p}}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $6$ and $2$ .

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Step 2.1.1

Factor $2$ out of $6 p$ .

$\frac{2 (3 p)}{2 p^{3} \sqrt{p}}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $2 p^{3} \sqrt{p}$ .

$\frac{2 (3 p)}{2 (p^{3} \sqrt{p})}$

Step 2.1.2.2

Cancel the common factor.

$\frac{2 (3 p)}{2 (p^{3} \sqrt{p})}$

Step 2.1.2.3

Rewrite the expression.

$\frac{3 p}{p^{3} \sqrt{p}}$

Step 2.2

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

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Step 2.2.1

Factor $p$ out of $3 p$ .

$\frac{p \cdot 3}{p^{3} \sqrt{p}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $p$ out of $p^{3} \sqrt{p}$ .

$\frac{p \cdot 3}{p (p^{2} \sqrt{p})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{p \cdot 3}{p (p^{2} \sqrt{p})}$

Step 2.2.2.3

Rewrite the expression.

$\frac{3}{p^{2} \sqrt{p}}$

Step 3

Multiply $\frac{3}{p^{2} \sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

$\frac{3}{p^{2} \sqrt{p}} \cdot \frac{\sqrt{p}}{\sqrt{p}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{3}{p^{2} \sqrt{p}}$ by $\frac{\sqrt{p}}{\sqrt{p}}$ .

$\frac{3 \sqrt{p}}{p^{2} \sqrt{p} \sqrt{p}}$

Step 4.2

Move $\sqrt{p}$ .

$\frac{3 \sqrt{p}}{p^{2} (\sqrt{p} \sqrt{p})}$

Step 4.3

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

$\frac{3 \sqrt{p}}{p^{2} ({\sqrt{p}}^{1} \sqrt{p})}$

Step 4.4

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

$\frac{3 \sqrt{p}}{p^{2} ({\sqrt{p}}^{1} {\sqrt{p}}^{1})}$

Step 4.5

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

$\frac{3 \sqrt{p}}{p^{2} {\sqrt{p}}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{3 \sqrt{p}}{p^{2} {\sqrt{p}}^{2}}$

Step 4.7

Rewrite ${\sqrt{p}}^{2}$ as $p$ .

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Step 4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{p}$ as $p^{\frac{1}{2}}$ .

$\frac{3 \sqrt{p}}{p^{2} {(p^{\frac{1}{2}})}^{2}}$

Step 4.7.2

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

$\frac{3 \sqrt{p}}{p^{2} p^{\frac{1}{2} \cdot 2}}$

Step 4.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \sqrt{p}}{p^{2} p^{\frac{2}{2}}}$

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{p}}{p^{2} p^{\frac{2}{2}}}$

Step 4.7.4.2

Rewrite the expression.

$\frac{3 \sqrt{p}}{p^{2} p^{1}}$

Step 4.7.5

Simplify.

$\frac{3 \sqrt{p}}{p^{2} p}$

Step 5

Multiply $p^{2}$ by $p$ by adding the exponents.

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Step 5.1

Multiply $p^{2}$ by $p$ .

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Step 5.1.1

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

$\frac{3 \sqrt{p}}{p^{2} p^{1}}$

Step 5.1.2

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

$\frac{3 \sqrt{p}}{p^{2 + 1}}$

Step 5.2

Add $2$ and $1$ .

$\frac{3 \sqrt{p}}{p^{3}}$