Algebra Examples

$\frac{\sqrt{5 n}}{2 \sqrt{15 n^{4}}}$

Step 1

Combine $\sqrt{5 n}$ and $\sqrt{15 n^{4}}$ into a single radical.

$\frac{\sqrt{\frac{5 n}{15 n^{4}}}}{2}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $5$ and $15$ .

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Step 2.1.1

Factor $5$ out of $5 n$ .

$\frac{\sqrt{\frac{5 (n)}{15 n^{4}}}}{2}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $5$ out of $15 n^{4}$ .

$\frac{\sqrt{\frac{5 (n)}{5 (3 n^{4})}}}{2}$

Step 2.1.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{5 n}{5 (3 n^{4})}}}{2}$

Step 2.1.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{n}{3 n^{4}}}}{2}$

Step 2.2

Cancel the common factor of $n$ and $n^{4}$ .

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Step 2.2.1

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

$\frac{\sqrt{\frac{n^{1}}{3 n^{4}}}}{2}$

Step 2.2.2

Factor $n$ out of $n^{1}$ .

$\frac{\sqrt{\frac{n \cdot 1}{3 n^{4}}}}{2}$

Step 2.2.3

Cancel the common factors.

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Step 2.2.3.1

Factor $n$ out of $3 n^{4}$ .

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

Step 2.2.3.2

Cancel the common factor.

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

Step 2.2.3.3

Rewrite the expression.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\frac{1}{3 n^{3}}$ as ${(\frac{1}{n})}^{2} \frac{1}{3 n}$ .

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Step 3.1.1

Factor the perfect power $1^{2}$ out of $1$ .

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

Step 3.1.2

Factor the perfect power $n^{2}$ out of $3 n^{3}$ .

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

Step 3.1.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{n^{2} \cdot (3 n)}$ .

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

Step 3.2

Pull terms out from under the radical.

$\frac{\frac{1}{n} \sqrt{\frac{1}{3 n}}}{2}$

Step 3.3

Rewrite $\sqrt{\frac{1}{3 n}}$ as $\frac{\sqrt{1}}{\sqrt{3 n}}$ .

$\frac{\frac{1}{n} \cdot \frac{\sqrt{1}}{\sqrt{3 n}}}{2}$

Step 3.4

Any root of $1$ is $1$ .

$\frac{\frac{1}{n} \cdot \frac{1}{\sqrt{3 n}}}{2}$

Step 3.5

Multiply $\frac{1}{\sqrt{3 n}}$ by $\frac{\sqrt{3 n}}{\sqrt{3 n}}$ .

$\frac{\frac{1}{n} (\frac{1}{\sqrt{3 n}} \cdot \frac{\sqrt{3 n}}{\sqrt{3 n}})}{2}$

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{1}{\sqrt{3 n}}$ by $\frac{\sqrt{3 n}}{\sqrt{3 n}}$ .

$\frac{\frac{1}{n} \cdot \frac{\sqrt{3 n}}{\sqrt{3 n} \sqrt{3 n}}}{2}$

Step 3.6.2

Raise $\sqrt{3 n}$ to the power of $1$ .

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

Step 3.6.3

Raise $\sqrt{3 n}$ to the power of $1$ .

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

Step 3.6.4

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

$\frac{\frac{1}{n} \cdot \frac{\sqrt{3 n}}{{\sqrt{3 n}}^{1 + 1}}}{2}$

Step 3.6.5

Add $1$ and $1$ .

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

Step 3.6.6

Rewrite ${\sqrt{3 n}}^{2}$ as $3 n$ .

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Step 3.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 n}$ as ${(3 n)}^{\frac{1}{2}}$ .

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

Step 3.6.6.2

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

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

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $2$ .

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Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

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

Step 3.6.6.5

Simplify.

$\frac{\frac{1}{n} \cdot \frac{\sqrt{3 n}}{3 n}}{2}$

Step 3.7

Multiply $\frac{1}{n} \cdot \frac{\sqrt{3 n}}{3 n}$ .

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Step 3.7.1

Multiply $\frac{1}{n}$ by $\frac{\sqrt{3 n}}{3 n}$ .

$\frac{\frac{\sqrt{3 n}}{n (3 n)}}{2}$

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.7.4

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

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

Step 3.7.5

Add $1$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply.

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Step 6.1

Multiply $\sqrt{3 n}$ by $1$ .

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

Step 6.2

Multiply $2$ by $3$ .

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