Question
Express the complex number ${\left( {\frac{1}{3} + 3i} \right)^3}$ in the form of $a + ib.$
✓
Answer
$\left(\frac13+3i\right)^3\;=\left(\frac13\right)^3+{(3i)^3}+3\times\left(\frac13\right)(3i)\left(\frac13+3i\right)$
$=\frac1{27}+27i^3+i+3i\left(\frac13+3i\right)$
$=\frac1{27}+27(-i)+i+9i^2$
$\left[ {\because {i^3} = - i\;and\;{i^2} = - 1} \right]$
$ =\frac1{27}-27i+i-9$
$=\left(\frac1{27}-9\right)-26i\;$
$=\frac{-242}{27}-26i$
$=\frac1{27}+27i^3+i+3i\left(\frac13+3i\right)$
$=\frac1{27}+27(-i)+i+9i^2$
$\left[ {\because {i^3} = - i\;and\;{i^2} = - 1} \right]$
$ =\frac1{27}-27i+i-9$
$=\left(\frac1{27}-9\right)-26i\;$
$=\frac{-242}{27}-26i$
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