Finite Math Examples

$\frac{21 \sqrt{6}}{84 \sqrt{30}}$

Step 1

Cancel the common factor of $21$ and $84$ .

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Step 1.1

Factor $21$ out of $21 \sqrt{6}$ .

$\frac{21 (\sqrt{6})}{84 \sqrt{30}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $21$ out of $84 \sqrt{30}$ .

$\frac{21 (\sqrt{6})}{21 (4 \sqrt{30})}$

Step 1.2.2

Cancel the common factor.

$\frac{21 \sqrt{6}}{21 (4 \sqrt{30})}$

Step 1.2.3

Rewrite the expression.

$\frac{\sqrt{6}}{4 \sqrt{30}}$

Step 2

Combine $\sqrt{6}$ and $\sqrt{30}$ into a single radical.

$\frac{\sqrt{\frac{6}{30}}}{4}$

Step 3

Cancel the common factor of $6$ and $30$ .

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Step 3.1

Factor $6$ out of $6$ .

$\frac{\sqrt{\frac{6 (1)}{30}}}{4}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $6$ out of $30$ .

$\frac{\sqrt{\frac{6 \cdot 1}{6 \cdot 5}}}{4}$

Step 3.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{6 \cdot 1}{6 \cdot 5}}}{4}$

Step 3.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{1}{5}}}{4}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$\frac{\frac{\sqrt{1}}{\sqrt{5}}}{4}$

Step 4.2

Any root of $1$ is $1$ .

$\frac{\frac{1}{\sqrt{5}}}{4}$

Step 4.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}{4}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}}{4}$

Step 4.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}{4}$

Step 4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}{4}$

Step 4.4.4

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

$\frac{\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}}{4}$

Step 4.4.5

Add $1$ and $1$ .

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

Step 4.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}}{4}$

Step 4.4.6.2

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

$\frac{\frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}}{4}$

Step 4.4.6.3

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

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

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

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

Step 4.4.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{5}}{5^{1}}}{4}$

Step 4.4.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{5}}{5}}{4}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{5}}{5} \cdot \frac{1}{4}$

Step 6

Multiply $\frac{\sqrt{5}}{5} \cdot \frac{1}{4}$ .

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Step 6.1

Multiply $\frac{\sqrt{5}}{5}$ by $\frac{1}{4}$ .

$\frac{\sqrt{5}}{5 \cdot 4}$

Step 6.2

Multiply $5$ by $4$ .

$\frac{\sqrt{5}}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{20}$

Decimal Form:

$0.11180339 \dots$