To simplify the expression [tex]\(i^{37}\)[/tex], we can use the properties of the imaginary unit [tex]\(i\)[/tex] and its powers. Recall the cyclic nature of the powers of [tex]\(i\)[/tex]:
[tex]\[
\begin{aligned}
i^1 &= i, \\
i^2 &= -1, \\
i^3 &= -i, \\
i^4 &= 1.
\end{aligned}
\][/tex]
Notice that [tex]\(i^4 = 1\)[/tex], which means any power of [tex]\(i\)[/tex] that is a multiple of [tex]\(4\)[/tex] will be equal to [tex]\(1\)[/tex]. Because of this, we can reduce the exponent [tex]\(37\)[/tex] modulo [tex]\(4\)[/tex] to simplify the expression.
Let's perform the steps:
1. Reduce the exponent modulo 4:
[tex]\[
37 \mod 4 = 1.
\][/tex]
This tells us that:
[tex]\[
i^{37} \equiv i^1 \pmod{4}.
\][/tex]
Thus:
[tex]\[
i^{37} = i^1.
\][/tex]
2. Identify the equivalent power of [tex]\(i\)[/tex]:
Based on the calculations above:
[tex]\[
i^{37} = i.
\][/tex]
So finally, the simplified expression for [tex]\(i^{37}\)[/tex] is:
[tex]\[
i^{37} = i.
\][/tex]
[tex]\(\boxed{i}\)[/tex]
Feel free to ask if you have any further questions regarding simplifying expressions that involve the imaginary unit [tex]\(i\)[/tex]!
