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Q)

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

$(a)\; \normalsize -\large\frac{342}{27} - \normalsize26i\qquad(b)\;\normalsize -\large\frac{242}{27} -\normalsize 26i\qquad(c)\; \normalsize -\large\frac{242}{27} -\normalsize 28i\qquad(d)\;\normalsize -\large\frac{242}{27} - \normalsize 25i$

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A)
Answer : $\;-\large\frac{242}{27} -26i$
Explanation :
$(\large\frac{1}{3}$$+3i)^{3} = (\large\frac{1}{3})^{3}+(3i)^{3}+ 3 (\large\frac{1}{3})(3i)(\large\frac{1}{3}+3i)$
$=\large\frac{1}{27}+27i^3+(3i) (\large\frac{1}{3}+3i) \qquad [i^{3} =-i]$
$=\large\frac{1}{27} +27(-i) +i+9i^2$
$= \large\frac{1}{27} -27i+i+9(-1) \qquad [i^2=-1]$
$=[\large\frac{1}{27} -9]+i(-27+1)$
$=-\large\frac{242}{27} -26i\;.$
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