Basic Math Examples

$\frac{\sqrt{5 c}}{\sqrt{15 c^{3}}}$

Step 1

Combine $\sqrt{5 c}$ and $\sqrt{15 c^{3}}$ into a single radical.

$\sqrt{\frac{5 c}{15 c^{3}}}$

Step 2

Reduce the expression $\frac{5 c}{15 c^{3}}$ by cancelling the common factors.

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Step 2.1

Factor $5$ out of $5 c$ .

$\sqrt{\frac{5 (c)}{15 c^{3}}}$

Step 2.2

Factor $5$ out of $15 c^{3}$ .

$\sqrt{\frac{5 (c)}{5 (3 c^{3})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{5 c}{5 (3 c^{3})}}$

Step 2.4

Rewrite the expression.

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

Step 3

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

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Step 3.1

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

$\sqrt{\frac{c^{1}}{3 c^{3}}}$

Step 3.2

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

$\sqrt{\frac{c \cdot 1}{3 c^{3}}}$

Step 3.3

Cancel the common factors.

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Step 3.3.1

Factor $c$ out of $3 c^{3}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Rearrange the fraction $\frac{1^{2} \cdot 1}{c^{2} \cdot 3}$ .

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

Step 5

Pull terms out from under the radical.

$\frac{1}{c} \sqrt{\frac{1}{3}}$

Step 6

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

$\frac{1}{c} \cdot \frac{\sqrt{1}}{\sqrt{3}}$

Step 7

Any root of $1$ is $1$ .

$\frac{1}{c} \cdot \frac{1}{\sqrt{3}}$

Step 8

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

$\frac{1}{c} (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

$\frac{1}{c} \cdot \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2

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

$\frac{1}{c} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.3

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

$\frac{1}{c} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.4

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

$\frac{1}{c} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

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

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Step 9.6.1

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

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

Step 9.6.2

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

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

Step 9.6.3

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

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

$\frac{1}{c} \cdot \frac{\sqrt{3}}{3^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{1}{c} \cdot \frac{\sqrt{3}}{3}$

Step 10

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

$\frac{\sqrt{3}}{c \cdot 3}$

Step 11

Move $3$ to the left of $c$ .

$\frac{\sqrt{3}}{3 c}$