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

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

Evaluate

$$\sqrt{3}+2\approx 3.732050808$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

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

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Subtract 1 from -3 to get -4.

$$\frac{-\sqrt{3}-4-\sqrt{3}}{-2}$$

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

$$\frac{-2\sqrt{3}-4}{-2}$$

Divide each term of -2\sqrt{3}-4 by -2 to get \sqrt{3}+2.

$$\sqrt{3}+2$$