Solve frac{2-isqrt{3}}{-sqrt{3}+i}

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

Evaluate

$$\frac{-\sqrt{3}i+2}{-\sqrt{3}+i}\approx -1.299038106+0.25i$$

Solution Steps

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

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

Consider \left(-\sqrt{3}+i\right)\left(-\sqrt{3}-i\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Calculate -\sqrt{3} to the power of 2 and get \left(\sqrt{3}\right)^{2}.

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

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

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

Multiply -1 and -1 to get 1.

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

The square of \sqrt{3} is 3.

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

Add 3 and 1 to get 4.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Subtract 2i from 3i to get i.

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

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

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

Real Part

$$\sqrt{3}Im(\frac{1}{-\sqrt{3}+i})+2Re(\frac{1}{-\sqrt{3}+i})$$