Question

Express the complex numbers in the form of a + ib:

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

Answer 1

${(-2-\frac{1}{3}i)}^3=-{(2+\frac{1}{3}i)}^3=-\left[\right(2{)}^3+{\left(\frac{1}{3}i\right)}^3+3\times (2{)}^2\frac{1}{3}i+3\times 2\times {\left(\frac{1}{3}i\right)}^2]$

$=-[8+\frac{1}{27}{i}^3+4i+\frac{2}{3}{i}^2]$

$=-[8-\frac{1}{27}i+4i-\frac{2}{3}]\left[\begin{array}{cc}\because & i^3=-i\\ & i^2=-1\end{array}\right]$

$=[(8-\frac{2}{3})+(4-\frac{1}{27})i]$

$=-[\frac{22}{3}+\frac{107}{27}i]=\frac{-22}{3}-\frac{107}{27}i$