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

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

Evaluate

$$-\frac{\sqrt{3}}{2}-\frac{3}{2}i\approx -0.866025404-1.5i$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 2i and 3 to get 6i.

$$\frac{\left(3\sqrt{3}-3i\right)\sqrt{3}}{6i}$$

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

$$\frac{3\left(\sqrt{3}\right)^{2}-3i\sqrt{3}}{6i}$$

The square of \sqrt{3} is 3.

$$\frac{3\times 3-3i\sqrt{3}}{6i}$$

Multiply 3 and 3 to get 9.

$$\frac{9-3i\sqrt{3}}{6i}$$

Real Part

$$-\frac{\sqrt{3}}{2} = -0.8660254037844386$$