Chapter 4.4, Problem 30PPS

To determine

Find the simplified form of the given expression.

Answer to Problem 30PPS

The simplified form of the expression is $5$ .

Explanation of Solution

Given:

The given expression is $\left(1+2i\right)\left(1-2i\right)$ .

Calculation:

Use FOIL method.

  $\begin{array}{l}\left(1+2i\right)\left(1-2i\right)\\ =1\cdot 1+1\cdot \left(-2i\right)+2i\left(1\right)+2i\left(-2i\right)\\ =1-2i+2i-4i^2\\ =1-4i^2\\ i^2=-1\\ =1-4\left(-1\right)\\ =1+4\\ =5\end{array}$

Or use identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$ .

  $\begin{array}{l}\left(1+2i\right)\left(1-2i\right)\\ =1-4i^2\\ i^2=-1\\ =1-4\left(-1\right)\\ =1+4\\ =5\end{array}$

Hence the simplified form of the expression is

  $5$ .