Express the complex number (-2-1 3 i )^3 in the form of a+i b. - Vidyadip

Question

Express the complex number $\left(-2-\frac{1}{3} i\right)^3$ in the form of $a+i b$.

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Answer

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

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