Question

Write (3-2i)^3 in simplest a + bi form.

Answer 1

We want to write

`\begin{gathered} \mleft(3-2i\mright)^3\text{ in simplest form } \\ a+bi \end{gathered}`

This means we have to expand

`(3-2i)^3`

Applying perfect cube formula, we have

`\begin{gathered} \mleft(a-b\mright)^3=a^3-3a^2b+3ab^2-b^3 \\ \text{where } \\ a=3,\: \: b=2i \end{gathered}`

We have

`\begin{gathered} (a-b)^3=a^3-3a^2b+3ab^2-b^3 \\ \mleft(3-2i\mright)^3=3^3-(3*3^2*2i)+(3*3*(2i)^2)-(2i)^3_{} \\ =27-(27*2i)+(9*(2i)^2)-(2i)^3_{} \end{gathered}`

This becomes

`\begin{gathered} \text{note that i = }\sqrt[]{-1} \\ i^2=\sqrt[]{-1^2}=-1 \\ So\text{ we have } \\ =27-(27*2i)+(9*(2i)^2)-(2i)^3_{} \\ 27-54i+(9*4i^2)-(8i^2* i) \\ 27-54i+(9*4*-1)-(8*-1* i) \\ 27-54i-36+8i \\ -9-46i \end{gathered}`

Hence the answer is

`-9-46i`