Algebra Examples

$\frac{4 \sqrt{2}}{\sqrt{20 p}}$

Step 1

Combine $\sqrt{2}$ and $\sqrt{20 p}$ into a single radical.

$4 \sqrt{\frac{2}{20 p}}$

Step 2

Reduce the expression $\frac{2}{20 p}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $2$ .

$4 \sqrt{\frac{2 (1)}{20 p}}$

Step 2.2

Factor $2$ out of $20 p$ .

$4 \sqrt{\frac{2 (1)}{2 (10 p)}}$

Step 2.3

Cancel the common factor.

$4 \sqrt{\frac{2 \cdot 1}{2 (10 p)}}$

Step 2.4

Rewrite the expression.

$4 \sqrt{\frac{1}{10 p}}$

Step 3

Rewrite $\sqrt{\frac{1}{10 p}}$ as $\frac{\sqrt{1}}{\sqrt{10 p}}$ .

$4 \frac{\sqrt{1}}{\sqrt{10 p}}$

Step 4

Any root of $1$ is $1$ .

$4 \frac{1}{\sqrt{10 p}}$

Step 5

Multiply $\frac{1}{\sqrt{10 p}}$ by $\frac{\sqrt{10 p}}{\sqrt{10 p}}$ .

$4 (\frac{1}{\sqrt{10 p}} \cdot \frac{\sqrt{10 p}}{\sqrt{10 p}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{1}{\sqrt{10 p}}$ by $\frac{\sqrt{10 p}}{\sqrt{10 p}}$ .

$4 \frac{\sqrt{10 p}}{\sqrt{10 p} \sqrt{10 p}}$

Step 6.2

Raise $\sqrt{10 p}$ to the power of $1$ .

$4 \frac{\sqrt{10 p}}{{\sqrt{10 p}}^{1} \sqrt{10 p}}$

Step 6.3

Raise $\sqrt{10 p}$ to the power of $1$ .

$4 \frac{\sqrt{10 p}}{{\sqrt{10 p}}^{1} {\sqrt{10 p}}^{1}}$

Step 6.4

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

$4 \frac{\sqrt{10 p}}{{\sqrt{10 p}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$4 \frac{\sqrt{10 p}}{{\sqrt{10 p}}^{2}}$

Step 6.6

Rewrite ${\sqrt{10 p}}^{2}$ as $10 p$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 p}$ as ${(10 p)}^{\frac{1}{2}}$ .

$4 \frac{\sqrt{10 p}}{{({(10 p)}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$4 \frac{\sqrt{10 p}}{{(10 p)}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$4 \frac{\sqrt{10 p}}{{(10 p)}^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$4 \frac{\sqrt{10 p}}{{(10 p)}^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$4 \frac{\sqrt{10 p}}{{(10 p)}^{1}}$

Step 6.6.5

Simplify.

$4 \frac{\sqrt{10 p}}{10 p}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $4$ .

$2 (2) \frac{\sqrt{10 p}}{10 p}$

Step 7.2

Factor $2$ out of $10 p$ .

$2 (2) \frac{\sqrt{10 p}}{2 (5 p)}$

Step 7.3

Cancel the common factor.

$2 \cdot 2 \frac{\sqrt{10 p}}{2 (5 p)}$

Step 7.4

Rewrite the expression.

$2 \frac{\sqrt{10 p}}{5 p}$

Step 8

Combine $2$ and $\frac{\sqrt{10 p}}{5 p}$ .

$\frac{2 \sqrt{10 p}}{5 p}$