Finite Math Examples

$\frac{4}{2 + \sqrt{3} + \sqrt{7}}$

Step 1

Multiply $\frac{4}{2 + \sqrt{3} + \sqrt{7}}$ by $\frac{2 + \sqrt{3} - \sqrt{7}}{2 + \sqrt{3} - \sqrt{7}}$ .

$\frac{4}{2 + \sqrt{3} + \sqrt{7}} \cdot \frac{2 + \sqrt{3} - \sqrt{7}}{2 + \sqrt{3} - \sqrt{7}}$

Step 2

Multiply $\frac{4}{2 + \sqrt{3} + \sqrt{7}}$ by $\frac{2 + \sqrt{3} - \sqrt{7}}{2 + \sqrt{3} - \sqrt{7}}$ .

$\frac{4 (2 + \sqrt{3} - \sqrt{7})}{(2 + \sqrt{3} + \sqrt{7}) (2 + \sqrt{3} - \sqrt{7})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{4 (2 + \sqrt{3} - \sqrt{7})}{4 + 2 \sqrt{3} - 2 \sqrt{7} + \sqrt{3} \cdot 2 + {\sqrt{3}}^{2} - \sqrt{21} + \sqrt{7} \cdot 2 + \sqrt{21} - {\sqrt{7}}^{2}}$

Step 4

Simplify.

$\frac{4 (2 + \sqrt{3} - \sqrt{7})}{4 \sqrt{3}}$

Step 5

Cancel the common factor of $4$ .

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Step 5.1

Cancel the common factor.

$\frac{4 (2 + \sqrt{3} - \sqrt{7})}{4 \sqrt{3}}$

Step 5.2

Rewrite the expression.

$\frac{2 + \sqrt{3} - \sqrt{7}}{\sqrt{3}}$

Step 6

Multiply $\frac{2 + \sqrt{3} - \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 + \sqrt{3} - \sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2 + \sqrt{3} - \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.3

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6

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

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Step 7.6.1

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{(2 + \sqrt{3} - \sqrt{7}) \sqrt{3}}{3}$

Decimal Form:

$0.62717530 \dots$