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

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

Evaluate

$$4$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Add 2 and 2 to get 4.

$$4-\sqrt{3}+\sqrt{3}$$

Combine -\sqrt{3} and \sqrt{3} to get 0.

$$4$$

Factor

$$2^{2}$$