Sure, let's determine the value of [tex]\(i^{82}\)[/tex].
1. Understanding the Imaginary Unit:
The imaginary unit [tex]\(i\)[/tex] is defined such that [tex]\(i^2 = -1\)[/tex].
2. Cycle of Powers of [tex]\(i\)[/tex]:
The powers of [tex]\(i\)[/tex] follow a cyclic pattern every four exponents:
[tex]\[
\begin{align*}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1, \\
i^5 & = i, \text{ and so on.}
\end{align*}
\][/tex]
This cycle repeats every four exponents.
3. Reducing the Exponent:
To find [tex]\(i^{82}\)[/tex], we need to reduce the exponent modulo 4 since the powers repeat every four steps.
[tex]\[
82 \mod 4 = 2
\][/tex]
This tells us that:
[tex]\[
i^{82} \equiv i^2 \mod 4
\][/tex]
4. Evaluating the Reduced Exponent:
From the cycle of powers of [tex]\(i\)[/tex], we know that [tex]\(i^2 = -1\)[/tex]. Therefore:
[tex]\[
i^{82} = -1
\][/tex]
So, the value of [tex]\(i^{82}\)[/tex] is:
[tex]\[
i^{82} = -1
\][/tex]
