Solve frac{sqrt{3}+2}{1-sqrt{3}}

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

Evaluate

$$\frac{-3\sqrt{3}-5}{2}\approx -5.098076211$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{3}+2}{1-\sqrt{3}} by multiplying numerator and denominator by 1+\sqrt{3}.

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

Consider \left(1-\sqrt{3}\right)\left(1+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Apply the distributive property by multiplying each term of \sqrt{3}+2 by each term of 1+\sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Add 3 and 2 to get 5.

$$\frac{\sqrt{3}+5+2\sqrt{3}}{-2}$$

Combine \sqrt{3} and 2\sqrt{3} to get 3\sqrt{3}.

$$\frac{3\sqrt{3}+5}{-2}$$

Multiply both numerator and denominator by -1.

$$\frac{-3\sqrt{3}-5}{2}$$