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

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

Evaluate

$$2$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

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

$$2$$

Factor

$$2$$