Finite Math Examples

$\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$

Step 1

Multiply $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ by $\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ .

$\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}} \cdot \frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$

Step 2

Multiply $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ by $\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ .

$\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{(\sqrt{2} + \sqrt{3} + \sqrt{5}) (\sqrt{2} + \sqrt{3} - \sqrt{5})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{{\sqrt{2}}^{2} + \sqrt{6} - \sqrt{10} + \sqrt{6} + {\sqrt{3}}^{2} - \sqrt{15} + \sqrt{10} + \sqrt{15} - {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2 \sqrt{6}}$

Step 5

Multiply $\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 6.2

Move $\sqrt{6}$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 6.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 6.4

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 6.5

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

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 6.6

Add $1$ and $1$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 6.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 6.7.2

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

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 6.7.3

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

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 6.7.4

Cancel the common factor of $2$ .

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Step 6.7.4.1

Cancel the common factor.

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 6.7.4.2

Rewrite the expression.

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \cdot 6^{1}}$

Step 6.7.5

Evaluate the exponent.

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{2 \cdot 6}$

Step 7

Multiply $2$ by $6$ .

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{12}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{(\sqrt{2} + \sqrt{3} - \sqrt{5}) \sqrt{6}}{12}$

Decimal Form:

$0.18579306 \dots$