To simplify [tex]\( i^{22} \)[/tex], we can use the properties of the imaginary unit [tex]\( i \)[/tex]. The imaginary unit [tex]\( i \)[/tex] has the following cyclical powers:
[tex]\[
\begin{align*}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1.
\end{align*}
\][/tex]
Notice that these powers repeat every 4 powers. This cycle ([tex]\( i, -1, -i, 1 \)[/tex]) is essential for simplifying higher powers of [tex]\( i \)[/tex].
To simplify [tex]\( i^{22} \)[/tex]:
1. Determine the position in the cycle:
Since the powers of [tex]\( i \)[/tex] repeat every 4 numbers, we find the position of [tex]\( i^{22} \)[/tex] in the cycle by computing the remainder when 22 is divided by 4.
[tex]\[
22 \div 4 = 5 \quad \text{ (quotient is not relevant here) }
\][/tex]
[tex]\[
22 \mod 4 = 2
\][/tex]
2. Identify the equivalent power:
The remainder is 2, which tells us that [tex]\( i^{22} \)[/tex] is equivalent to [tex]\( i^2 \)[/tex] based on the cyclical pattern.
3. Use the known value from the cycle:
From the properties of [tex]\( i \)[/tex]:
[tex]\[
i^2 = -1
\][/tex]
Hence, simplifying [tex]\( i^{22} \)[/tex] gives us:
[tex]\[
i^{22} = -1
\][/tex]
