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)

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