Basic Math Examples

$\frac{1}{3 + \sqrt{7}} + \frac{1}{3 - \sqrt{7}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

$\frac{1}{3 + \sqrt{7}} \cdot \frac{3 - \sqrt{7}}{3 - \sqrt{7}} + \frac{1}{3 - \sqrt{7}}$

Step 1.2

Multiply $\frac{1}{3 + \sqrt{7}}$ by $\frac{3 - \sqrt{7}}{3 - \sqrt{7}}$ .

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

Step 1.3

Expand the denominator using the FOIL method.

$\frac{3 - \sqrt{7}}{9 - 3 \sqrt{7} + \sqrt{7} \cdot 3 - {\sqrt{7}}^{2}} + \frac{1}{3 - \sqrt{7}}$

Step 1.4

Simplify.

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

Step 1.5

Multiply $\frac{1}{3 - \sqrt{7}}$ by $\frac{3 + \sqrt{7}}{3 + \sqrt{7}}$ .

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

Step 1.6

Multiply $\frac{1}{3 - \sqrt{7}}$ by $\frac{3 + \sqrt{7}}{3 + \sqrt{7}}$ .

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

Step 1.7

Expand the denominator using the FOIL method.

$\frac{3 - \sqrt{7}}{2} + \frac{3 + \sqrt{7}}{9 + 3 \sqrt{7} - 3 \sqrt{7} - {\sqrt{7}}^{2}}$

Step 1.8

Simplify.

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Add $3$ and $3$ .

$\frac{6 - \sqrt{7} + \sqrt{7}}{2}$

Step 2.3

Add $- \sqrt{7}$ and $\sqrt{7}$ .

$\frac{6 + 0}{2}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $6$ and $0$ .

$\frac{6}{2}$

Step 2.4.2

Divide $6$ by $2$ .

$3$