Question

Express the complex numbers in the form of a + ib:

${(\frac{1}{3}+3i)}^3$

Answer 1

${(\frac{1}{3}+3i)}^3={\left(\frac{1}{3}\right)}^3+(3i{)}^3+3\times {\left(\frac{1}{3}\right)}^2\times 3i+3\times \frac{1}{3}\times (3i{)}^2$

$=\frac{1}{27}+27i^3+i+9i^2$

$=\frac{1}{27}-27i+i-9$

$[\because i^3=-i\text{and}{i}^2=-1]$

$=(\frac{1}{27}-9)-26i=\frac{-242}{27}-26i$