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

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

Evaluate

$$3-2\sqrt{3}\approx -0.464101615$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Add 2 and 3 to get 5.

$$5-\sqrt{3}-\sqrt{3}-2$$

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

$$5-2\sqrt{3}-2$$

Subtract 2 from 5 to get 3.

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