Question

Express the given complex number in the form $a+ib:$

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

Answer 1

${(-2-\frac{1}{3}i)}^3={(-1)}^3{(2+\frac{1}{3}i)}^3$
$=-[2^3+{\left(\frac{i}{3}\right)}^3+3\left(2\right)\left(\frac{i}{3}\right)(2+\frac{i}{3})]$
$=-[8+\frac{i^3}{27}+2i(2+\frac{i}{3})]$
$=-(8-\frac{i}{27}+4i+\frac{2i^2}{3})$ $(\because i^3=-i)$
$=-(8-\frac{i}{27}+4i-\frac{2}{3})$ $(\because i^2=-1)$
$=-(\frac{22}{3}+\frac{107i}{27})$
$=-\frac{22}{3}-\frac{107}{27}i$