Question

Express $\left(\frac{1}{3}+3i\right)^{3}$ in the form of $a +ib$.

• A. $1+26i$
• B. $\frac{-242}{27}+26i$
• C. $\frac{-242}{27}-26i$
• D. $-1+26i$

Answer 1

Answer: (C)

$\left(\frac{1}{3}+3i\right)^{3} =\left(\frac{1}{3}\right)^{3}+3\left(\frac{1}{3}\right)^{2} \left(3i\right)+3\left(\frac{1}{3}\right)\left(3i\right)^{2}+\left(3i\right)^{3}$
$=\frac{1}{27}+i+9\left(-1\right)+27\left(-i\right)$
$=\frac{1}{27}-9-26i$
$=\frac{1-243}{27}-26i$
$=\frac{-242}{27}-26i$