To find the value of [tex]\( i^{213} \)[/tex], we need to consider the properties of the imaginary unit [tex]\( i \)[/tex]. The powers of [tex]\( i \)[/tex] follow a specific pattern that repeats every four exponents. Here is the pattern for [tex]\( i \)[/tex]:
[tex]\[
\begin{align*}
i^1 &= i \\
i^2 &= -1 \\
i^3 &= -i \\
i^4 &= 1 \\
\end{align*}
\][/tex]
This pattern repeats for higher powers of [tex]\( i \)[/tex]. Therefore, to determine [tex]\( i^{213} \)[/tex], we only need to find the remainder when 213 is divided by the cycle length, which is 4.
Let's compute the remainder:
[tex]\[
213 \div 4 = 53 \text{ R } 1
\][/tex]
The remainder is 1, so [tex]\( i^{213} \)[/tex] has the same value as [tex]\( i^1 \)[/tex]. According to the pattern:
[tex]\[
i^1 = i
\][/tex]
Therefore, the value of [tex]\( i^{213} \)[/tex] is:
[tex]\[
\boxed{i}
\][/tex]
