Chapter 9, Problem 48RE

To determine

To write: Complex number ${\left(1-2i\right)}^4$ in the standard form of $\left(a+ib\right)$ .

Answer to Problem 48RE

The mentioned complex number in the standard form is $-7+i\left(24\right)$ .

Explanation of Solution

Given information:

The mentioned complex number is ${\left(1-2i\right)}^4$ .

Formula used:

By the use of binomial expansion ${\left(x-y\right)}^2=x^2-2xy+y^2$ .

Calculation:

Consider the mentioned complex number is ${\left(1-2i\right)}^4$ .

Recall that binomial expansion of ${\left(x-y\right)}^2=x^2-2xy+y^2$ .

  $\begin{array}{l}{\left(1-2i\right)}^2=1^2-2*1*2i+{\left(2i\right)}^2=1-4i+4i^2=1-4i-4=-3-4i=-\left(3+4i\right)\\ \Rightarrow {\left(1-2i\right)}^4={\left(-\left(3+4i\right)\right)}^2=9+24i+16i^2=9-16+24i=-7+24i\end{array}$

Since ${\left(a-bi\right)}^4={\left({\left(a-bi\right)}^2\right)}^2$ .Therefore we evaluate ${\left(a-bi\right)}^2$ first.

Thus the required Complex number ${\left(1-2i\right)}^4$ in the standard form of $\left(a+bi\right)$ is $-7+i\left(24\right)$ .