Solve 1-sqrt{3}-1/2-sqrt{3}

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

Evaluate

$$-2\sqrt{3}-1\approx -4.464101615$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

To find the opposite of 2+\sqrt{3}, find the opposite of each term.

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

Subtract 2 from 1 to get -1.

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

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

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