Solve 2/-1+sqrt{3i}

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

Evaluate

$$\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

Solution Steps

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

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

Consider \left(-1+i\sqrt{3}\right)\left(-1-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}.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply -1 and -3 to get 3.

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

Add 1 and 3 to get 4.

$$\frac{2\left(-1-i\sqrt{3}\right)}{4}$$

Divide 2\left(-1-i\sqrt{3}\right) by 4 to get \frac{1}{2}\left(-1-i\sqrt{3}\right).

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

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

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Multiply \frac{1}{2} and -i to get -\frac{1}{2}i.

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

Real Part

$$-\frac{1}{2} = -0.5$$