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Complex Numbers and Quadratic Equations — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers and Quadratic Equations2 Marks
Question
Express the complex number ${\left( { - 2 - \frac{1}{3}i} \right)^3}$ in the form of a + ib.
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Answer

${\left( { - 2 - \frac{1}{3}i} \right)^3}$$ = - {\left( {2 + \frac{1}{3}i} \right)^3}$
$=-\left[{(2)}^3+\left(\frac13i\right)^3+3\times{(2)}^2\times\frac13i\right.$$\left. { + 3 \times 2 \times {{\left( {\frac{1}{3}i} \right)}^2}} \right]$
$=-\left[8+\frac1{27}i^3+4i+\frac23i^2\right]$$ = - \left[ {8 - \frac{1}{{27}}i + 4i - \frac{2}{3}} \right]\left[ {\begin{array}{*{20}{c}} {\because {i^3} = - i} \\ {{i^2} = - 1} \end{array}} \right]$
$ = \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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