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

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

Evaluate

$$2-\sqrt{3}\approx 0.267949192$$

Solution Steps

Divide 2-\sqrt{3} by 2-\sqrt{3} to get 1.

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

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

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

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}.

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

Square 2. Square \sqrt{3}.

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

Subtract 3 from 4 to get 1.

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

Anything divided by one gives itself.

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

Use the distributive property to multiply 2-\sqrt{3} by 1.

$$2-\sqrt{3}$$