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

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

Evaluate

$$-3\sqrt{3}-5\approx -10.196152423$$

Solution Steps

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

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

Consider \left(2-\sqrt{3}\right)\left(2+\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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Subtract 3 from -2 to get -5.

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