Chapter 12.3, Problem 5E

To determine

To simplify: expression $i^{19}$ to $i,-1,-i,or1$

Answer to Problem 5E

The required answer is: $-i$

Explanation of Solution

Given information:

The given expression: $i^{19}$

Calculation:

  $\begin{array}{l}{i}^1=\sqrt{-1}=i\\ $ i^2={(\sqrt{-1})}^2=-1$i^3=i\cdot i^2=i(-1)=-i\\ i^4=i^2\cdot i^2=(-1)(-1)=1\end{array}$

Now,

  $i^{19}=i^8\cdot i^8\cdot i^3$

(Product of powers property)

  $=(1)(1)(-i)$

(Since $i^8=i^4\cdot i^4=(1)(1)=1$ )

  $=-i$

(Simplify)