Algebra Examples

$\frac{\frac{1}{13}}{1 - \frac{\sqrt{170}}{13}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{13} \cdot \frac{1}{1 - \frac{\sqrt{170}}{13}}$

Step 2

Simplify the denominator.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\frac{1}{13} \cdot \frac{1}{\frac{13}{13} - \frac{\sqrt{170}}{13}}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{1}{13} \cdot \frac{1}{\frac{13 - \sqrt{170}}{13}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{13} (1 \frac{13}{13 - \sqrt{170}})$

Step 4

Multiply $\frac{13}{13 - \sqrt{170}}$ by $1$ .

$\frac{1}{13} \cdot \frac{13}{13 - \sqrt{170}}$

Step 5

Cancel the common factor of $13$ .

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Step 5.1

Cancel the common factor.

$\frac{1}{13} \cdot \frac{13}{13 - \sqrt{170}}$

Step 5.2

Rewrite the expression.

$\frac{1}{13 - \sqrt{170}}$

Step 6

Multiply $\frac{1}{13 - \sqrt{170}}$ by $\frac{13 + \sqrt{170}}{13 + \sqrt{170}}$ .

$\frac{1}{13 - \sqrt{170}} \cdot \frac{13 + \sqrt{170}}{13 + \sqrt{170}}$

Step 7

Multiply $\frac{1}{13 - \sqrt{170}}$ by $\frac{13 + \sqrt{170}}{13 + \sqrt{170}}$ .

$\frac{13 + \sqrt{170}}{(13 - \sqrt{170}) (13 + \sqrt{170})}$

Step 8

Expand the denominator using the FOIL method.

$\frac{13 + \sqrt{170}}{169 + 13 \sqrt{170} - 13 \sqrt{170} - {\sqrt{170}}^{2}}$

Step 9

Simplify.

$\frac{13 + \sqrt{170}}{- 1}$

Step 10

Simplify the expression.

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Step 10.1

Move the negative one from the denominator of $\frac{13 + \sqrt{170}}{- 1}$ .

$- 1 \cdot (13 + \sqrt{170})$

Step 10.2

Rewrite $- 1 \cdot (13 + \sqrt{170})$ as $- (13 + \sqrt{170})$ .

$- (13 + \sqrt{170})$

Step 11

Apply the distributive property.

$- 1 \cdot 13 - \sqrt{170}$

Step 12

Multiply $- 1$ by $13$ .

$- 13 - \sqrt{170}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- 13 - \sqrt{170}$

Decimal Form:

$- 26.03840481 \dots$