Solve [-sqrt{3}+2/15i][18sqrt{3}-4/5i]

$$[ - \sqrt { 3 } + \frac { 2 } { 15 } i ] [ 18 \sqrt { 3 } - \frac { 4 } { 5 } i ]$$

Evaluate

$$\frac{16\sqrt{3}i}{5}-\frac{4042}{75}\approx -53.893333333+5.542562584i$$

Solution Steps

Apply the distributive property by multiplying each term of -\sqrt{3}+\frac{2}{15}i by each term of 18\sqrt{3}-\frac{4}{5}i.

$$18\left(-\sqrt{3}\right)\sqrt{3}-\frac{4}{5}i\left(-\sqrt{3}\right)+\frac{12}{5}i\sqrt{3}+\frac{8}{75}$$

Multiply 18 and -1 to get -18.

$$-18\sqrt{3}\sqrt{3}-\frac{4}{5}i\left(-1\right)\sqrt{3}+\frac{12}{5}i\sqrt{3}+\frac{8}{75}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$-18\times 3-\frac{4}{5}i\left(-1\right)\sqrt{3}+\frac{12}{5}i\sqrt{3}+\frac{8}{75}$$

Multiply -18 and 3 to get -54.

$$-54-\frac{4}{5}i\left(-1\right)\sqrt{3}+\frac{12}{5}i\sqrt{3}+\frac{8}{75}$$

Multiply -\frac{4}{5}i and -1 to get \frac{4}{5}i.

$$-54+\frac{4}{5}i\sqrt{3}+\frac{12}{5}i\sqrt{3}+\frac{8}{75}$$

Combine \frac{4}{5}i\sqrt{3} and \frac{12}{5}i\sqrt{3} to get \frac{16}{5}i\sqrt{3}.

$$-54+\frac{16}{5}i\sqrt{3}+\frac{8}{75}$$

Convert -54 to fraction -\frac{4050}{75}.

$$-\frac{4050}{75}+\frac{16}{5}i\sqrt{3}+\frac{8}{75}$$

Since -\frac{4050}{75} and \frac{8}{75} have the same denominator, add them by adding their numerators.

$$\frac{-4050+8}{75}+\frac{16}{5}i\sqrt{3}$$

Add -4050 and 8 to get -4042.

$$-\frac{4042}{75}+\frac{16}{5}i\sqrt{3}$$

Real Part

$$-\frac{4042}{75} = -53\frac{67}{75} = -53.89333333333333$$