Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Expand the denominator using the FOIL method.

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

Step 1.4

Simplify.

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Multiply $- 1$ by $1$ .

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

Step 1.9

Multiply $\frac{1}{\sqrt{2} - \sqrt{3}}$ by $\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}}$ .

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

Step 1.10

Multiply $\frac{1}{\sqrt{2} - \sqrt{3}}$ by $\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}}$ .

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

Step 1.11

Expand the denominator using the FOIL method.

$- 1 - \sqrt{2} - \frac{\sqrt{2} + \sqrt{3}}{{\sqrt{2}}^{2} + \sqrt{6} - \sqrt{6} - {\sqrt{3}}^{2}} + \frac{1}{\sqrt{3} - 2}$

Step 1.12

Simplify.

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

Step 1.13

Move the negative one from the denominator of $\frac{\sqrt{2} + \sqrt{3}}{- 1}$ .

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

Step 1.14

Rewrite $- 1 \cdot (\sqrt{2} + \sqrt{3})$ as $- (\sqrt{2} + \sqrt{3})$ .

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

Step 1.15

Apply the distributive property.

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

Step 1.16

Apply the distributive property.

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

Step 1.17

Multiply $- - \sqrt{2}$ .

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Step 1.17.1

Multiply $- 1$ by $- 1$ .

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

Step 1.17.2

Multiply $\sqrt{2}$ by $1$ .

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

Step 1.18

Multiply $- - \sqrt{3}$ .

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Step 1.18.1

Multiply $- 1$ by $- 1$ .

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

Step 1.18.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 1.19

Multiply $\frac{1}{\sqrt{3} - 2}$ by $\frac{\sqrt{3} + 2}{\sqrt{3} + 2}$ .

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

Step 1.20

Multiply $\frac{1}{\sqrt{3} - 2}$ by $\frac{\sqrt{3} + 2}{\sqrt{3} + 2}$ .

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

Step 1.21

Expand the denominator using the FOIL method.

$- 1 - \sqrt{2} + \sqrt{2} + \sqrt{3} + \frac{\sqrt{3} + 2}{{\sqrt{3}}^{2} + \sqrt{3} \cdot 2 - 2 \sqrt{3} - 4}$

Step 1.22

Simplify.

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

Step 1.23

Move the negative one from the denominator of $\frac{\sqrt{3} + 2}{- 1}$ .

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

Step 1.24

Rewrite $- 1 \cdot (\sqrt{3} + 2)$ as $- (\sqrt{3} + 2)$ .

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

Step 1.25

Apply the distributive property.

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

Step 1.26

Multiply $- 1$ by $2$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $2$ from $- 1$ .

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

Step 2.2

Add $- \sqrt{2}$ and $\sqrt{2}$ .

$- 3 + 0 + \sqrt{3} - \sqrt{3}$

Step 2.3

Add $- 3$ and $0$ .

$- 3 + \sqrt{3} - \sqrt{3}$

Step 2.4

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

$- 3 + 0$

Step 2.5

Add $- 3$ and $0$ .

$- 3$