Q. 578 - Intermediate Algebra

Question

In the following exercises, simplify.

$i^{255}$

Step-by-Step Solution

Answer

The simplified form is $-i$.

Step 1. Given information.

The given imaginary number is $i^{255}$.

Step 2. Simplify the power.

We know that, 

$$\begin{gathered} i=\sqrt{-1} \\ {i}^2=-1 \\ {i}^3=-i \\ {i}^4=1 \end{gathered}$$

Divide the power by $4$ and rewrite,

$$\begin{gathered} i^{255}=i^{4·63+3} \\ ={\left(i^4\right)}^{63}·i^3 \\ ={\left(1\right)}^{63}·i^3 \\ =1·i^3 \\ =-i \end{gathered}$$

Therefore, the simplified form is $-i$