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

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

Evaluate

$$3$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Subtract 1 from 2 to get 1.

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

Add 1 and 2 to get 3.

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

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

$$3$$

Factor

$$3$$