Simplify 4/(1- square root of 3)

$\frac{4}{1 - \sqrt{3}}$

Step 1

Multiply $\frac{4}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

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

Step 2

Multiply $\frac{4}{1 - \sqrt{3}}$ by $\frac{1 + \sqrt{3}}{1 + \sqrt{3}}$ .

$\frac{4 (1 + \sqrt{3})}{(1 - \sqrt{3}) (1 + \sqrt{3})}$

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{4 (1 + \sqrt{3})}{- 2}$

Step 5

Cancel the common factor of $4$ and $- 2$ .

Factor $2$ out of $4 (1 + \sqrt{3})$ .

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

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

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

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $2$ by $1$ .

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

Step 9

Apply the distributive property.

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

Step 10

Multiply.

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $- 1$ .

$- 2 - 2 \sqrt{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 2 - 2 \sqrt{3}$

Decimal Form:

$- 5.46410161 \dots$