Basic Math Examples

$\frac{1}{10 - 6 \sqrt{2}} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{10 - 6 \sqrt{2}}$ by $\frac{10 + 6 \sqrt{2}}{10 + 6 \sqrt{2}}$ .

$\frac{1}{10 - 6 \sqrt{2}} \cdot \frac{10 + 6 \sqrt{2}}{10 + 6 \sqrt{2}} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.2

Multiply $\frac{1}{10 - 6 \sqrt{2}}$ by $\frac{10 + 6 \sqrt{2}}{10 + 6 \sqrt{2}}$ .

$\frac{10 + 6 \sqrt{2}}{(10 - 6 \sqrt{2}) (10 + 6 \sqrt{2})} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.3

Expand the denominator using the FOIL method.

$\frac{10 + 6 \sqrt{2}}{100 + 60 \sqrt{2} - 60 \sqrt{2} - 36 {\sqrt{2}}^{2}} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.4

Simplify.

$\frac{10 + 6 \sqrt{2}}{28} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5

Cancel the common factor of $10 + 6 \sqrt{2}$ and $28$ .

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Step 1.5.1

Factor $2$ out of $10$ .

$\frac{2 (5) + 6 \sqrt{2}}{28} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5.2

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{2 (5) + 2 (3 \sqrt{2})}{28} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5.3

Factor $2$ out of $2 (5) + 2 (3 \sqrt{2})$ .

$\frac{2 (5 + 3 \sqrt{2})}{28} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5.4

Cancel the common factors.

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Step 1.5.4.1

Factor $2$ out of $28$ .

$\frac{2 (5 + 3 \sqrt{2})}{2 \cdot 14} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5.4.2

Cancel the common factor.

$\frac{2 (5 + 3 \sqrt{2})}{2 \cdot 14} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.5.4.3

Rewrite the expression.

$\frac{5 + 3 \sqrt{2}}{14} + \frac{1}{10 + 6 \sqrt{2}}$

Step 1.6

Multiply $\frac{1}{10 + 6 \sqrt{2}}$ by $\frac{10 - 6 \sqrt{2}}{10 - 6 \sqrt{2}}$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{1}{10 + 6 \sqrt{2}} \cdot \frac{10 - 6 \sqrt{2}}{10 - 6 \sqrt{2}}$

Step 1.7

Multiply $\frac{1}{10 + 6 \sqrt{2}}$ by $\frac{10 - 6 \sqrt{2}}{10 - 6 \sqrt{2}}$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{10 - 6 \sqrt{2}}{(10 + 6 \sqrt{2}) (10 - 6 \sqrt{2})}$

Step 1.8

Expand the denominator using the FOIL method.

$\frac{5 + 3 \sqrt{2}}{14} + \frac{10 - 6 \sqrt{2}}{100 - 60 \sqrt{2} + 60 \sqrt{2} - 36 {\sqrt{2}}^{2}}$

Step 1.9

Simplify.

$\frac{5 + 3 \sqrt{2}}{14} + \frac{10 - 6 \sqrt{2}}{28}$

Step 1.10

Cancel the common factor of $10 - 6 \sqrt{2}$ and $28$ .

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Step 1.10.1

Factor $2$ out of $10$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{2 (5) - 6 \sqrt{2}}{28}$

Step 1.10.2

Factor $2$ out of $- 6 \sqrt{2}$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{2 (5) + 2 (- 3 \sqrt{2})}{28}$

Step 1.10.3

Factor $2$ out of $2 (5) + 2 (- 3 \sqrt{2})$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{2 (5 - 3 \sqrt{2})}{28}$

Step 1.10.4

Cancel the common factors.

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Step 1.10.4.1

Factor $2$ out of $28$ .

$\frac{5 + 3 \sqrt{2}}{14} + \frac{2 (5 - 3 \sqrt{2})}{2 \cdot 14}$

Step 1.10.4.2

Cancel the common factor.

$\frac{5 + 3 \sqrt{2}}{14} + \frac{2 (5 - 3 \sqrt{2})}{2 \cdot 14}$

Step 1.10.4.3

Rewrite the expression.

$\frac{5 + 3 \sqrt{2}}{14} + \frac{5 - 3 \sqrt{2}}{14}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{5 + 3 \sqrt{2} + 5 - 3 \sqrt{2}}{14}$

Step 2.2

Add $5$ and $5$ .

$\frac{10 + 3 \sqrt{2} - 3 \sqrt{2}}{14}$

Step 2.3

Subtract $3 \sqrt{2}$ from $3 \sqrt{2}$ .

$\frac{10 + 0}{14}$

Step 2.4

Add $10$ and $0$ .

$\frac{10}{14}$

Step 2.5

Cancel the common factor of $10$ and $14$ .

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Step 2.5.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{14}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $14$ .

$\frac{2 \cdot 5}{2 \cdot 7}$

Step 2.5.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 7}$

Step 2.5.2.3

Rewrite the expression.

$\frac{5}{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{7}$

Decimal Form:

$0.\overline{714285}$