Basic Math Examples

$\frac{1}{1 - \sqrt{2}} - \frac{1}{\sqrt{2 - 1}}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

$\frac{1 + \sqrt{2}}{1 + \sqrt{2} - \sqrt{2} - {\sqrt{2}}^{2}} - \frac{1}{\sqrt{2 - 1}}$

Step 1.4

Simplify.

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

Step 1.5

Move the negative one from the denominator of $\frac{1 + \sqrt{2}}{- 1}$ .

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

Step 1.6

Rewrite $- 1 \cdot (1 + \sqrt{2})$ as $- (1 + \sqrt{2})$ .

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

Step 1.7

Apply the distributive property.

$- 1 \cdot 1 - \sqrt{2} - \frac{1}{\sqrt{2 - 1}}$

Step 1.8

Multiply $- 1$ by $1$ .

$- 1 - \sqrt{2} - \frac{1}{\sqrt{2 - 1}}$

Step 1.9

Simplify the denominator.

Tap for more steps...

Step 1.9.1

Subtract $1$ from $2$ .

$- 1 - \sqrt{2} - \frac{1}{\sqrt{1}}$

Step 1.9.2

Any root of $1$ is $1$ .

$- 1 - \sqrt{2} - \frac{1}{1}$

Step 1.10

Cancel the common factor of $1$ .

Tap for more steps...

Step 1.10.1

Cancel the common factor.

$- 1 - \sqrt{2} - \frac{1}{1}$

Step 1.10.2

Rewrite the expression.

$- 1 - \sqrt{2} - 1 \cdot 1$

Step 1.11

Multiply $- 1$ by $1$ .

$- 1 - \sqrt{2} - 1$

Step 2

Subtract $1$ from $- 1$ .

$- 2 - \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{2}$

Decimal Form:

$- 3.41421356 \dots$