Solve 1/2+sqrt{3i}

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

Evaluate

$$\frac{-\sqrt{3}i+2}{7}\approx 0.285714286-0.24743583i$$

Solution Steps

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

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

Consider \left(2+i\sqrt{3}\right)\left(2-i\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}.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{2-i\sqrt{3}}{4-\left(i\sqrt{3}\right)^{2}}$$

Expand \left(i\sqrt{3}\right)^{2}.

$$\frac{2-i\sqrt{3}}{4-i^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate i to the power of 2 and get -1.

$$\frac{2-i\sqrt{3}}{4-\left(-\left(\sqrt{3}\right)^{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{2-i\sqrt{3}}{4-\left(-3\right)}$$

Multiply -1 and -3 to get 3.

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

Add 4 and 3 to get 7.

$$\frac{2-i\sqrt{3}}{7}$$

Real Part

$$\frac{2}{7} = 0.2857142857142857$$