Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

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

Step 5

Apply the distributive property.

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

Step 6

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

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

Step 7

Combine using the product rule for radicals.

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

Step 8

Simplify the expression.

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Step 8.1

Multiply $2$ by $3$ .

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

Step 8.2

Move the negative in front of the fraction.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.93185165 \dots$