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Q)

Express the given complex number in the form $\;a+ib : (-2-\large\frac{1}{3} \normalsize i)^{3}$

$(a)\;-\large\frac{107}{3} \normalsize-\large\frac{22}{27} \normalsize i\qquad(b)\;-\large\frac{107}{27} \normalsize-\large\frac{22}{3} i \qquad(c)\;-\large\frac{22}{3} \normalsize-\large\frac{107}{27} \normalsize i\qquad(d)\;-\large\frac{11}{3} \normalsize-\large\frac{107}{27} \normalsize i$

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A)
Answer : $\;-\large\frac{22}{3} \normalsize-\large\frac{107}{27} \normalsize i$
Explanation :
$(-2-\large\frac{1}{3} \normalsize i)^{3} = (-1)^{3}\;(1 \normalsize+\large\frac{1}{3} \normalsize i)$
$=-[2^{3} +(\large\frac{i}{3})^{3} + \normalsize (3) \normalsize (2) \normalsize (\large\frac{i}{3})(\normalsize 2+\large\frac{i}{3})]$
$=-[8+\large\frac{i^3}{27}+\normalsize 2i(2+\large\frac{i}{3})]$
$=-[8-\large\frac{i}{27} + \normalsize 4i +\large\frac{2i^2}{3}] $
$=-\large\frac{22}{3} - \large\frac{107}{27} \normalsize i$
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