To determine the value of [tex]\(i^{233}\)[/tex], let's analyze the behavior of the imaginary unit [tex]\(i\)[/tex]. The imaginary unit [tex]\(i\)[/tex] has the property that [tex]\(i^2 = -1\)[/tex]. Additionally, the powers of [tex]\(i\)[/tex] follow a repeating pattern every four steps:
[tex]\[
\begin{aligned}
i^1 &= i, \\
i^2 &= -1, \\
i^3 &= -i, \\
i^4 &= 1.
\end{aligned}
\][/tex]
This cycle repeats every four powers. Therefore, any power of [tex]\(i\)[/tex] can be reduced by finding the remainder when the exponent is divided by 4.
Let's perform this operation with the exponent 233:
[tex]\[
233 \div 4 = 58 \text{ remainder } 1
\][/tex]
So, we can write 233 as:
[tex]\[
233 = 4 \times 58 + 1
\][/tex]
This tells us that:
[tex]\[
i^{233} \equiv i^1 \pmod{4}
\][/tex]
From our known pattern, [tex]\(i^1 = i\)[/tex].
Thus, [tex]\(i^{233} = i\)[/tex].
So, the expression equivalent to [tex]\(i^{233}\)[/tex] is:
[tex]\[
\boxed{i}
\][/tex]
