Chapter 6.7, Problem 53WE

To determine

To Find:

Simplify $i^n;n=1,2,3,\mathrm{...},12$ .

Find the pattern.

Answer to Problem 53WE

The pattern repeats after every 4^th power

  $\{1,i,-1,-i\}$

Explanation of Solution

Concept Used:

  $i=\sqrt{-1}$

Calculation:

So,

  $\begin{array}{l}{i}^1=\sqrt{-1}\\ i^2={\left(\sqrt{-1}\right)}^2=-1\\ i^3=i\cdot i^2=i(-1)=-i\\ i^4=i^2\cdot i^2=(-1)(-1)=1\\ i^5=i\cdot i^4=i(1)=i\\ i^6=i^2\cdot i^4=-1\\ .\\ .\\ .\\ i^{12}=i^{4(3)}=1\end{array}$

So, we can observe the pattern :

  $\begin{array}{l}{i}^{4n}=1\\ i^{4n+1}=i\\ i^{4n+2}=-1\\ i^{4n+3}=-i\end{array}$

where n is whole number

Hence the pattern is that it repeats after every 4^th power in the pattern :

  $\{1,i,-1,-i\}$ , for the power multiple of 4 , the value is 1.