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

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

Evaluate

$$-\sqrt{3}-3\approx -4.732050808$$

Solution Steps

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

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

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}.

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

Square 1. Square \sqrt{3}.

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

Subtract 3 from 1 to get -2.

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

Use the distributive property to multiply 2\sqrt{3} by 1+\sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

Divide each term of 2\sqrt{3}+6 by -2 to get -\sqrt{3}-3.

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