Let's delve into solving the problem of finding [tex]\( i^{79} \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit defined by the property [tex]\( i^2 = -1 \)[/tex].
Here’s a step-by-step breakdown:
1. Understand the Powers of [tex]\( i \)[/tex]:
The powers of [tex]\( i \)[/tex] follow a cyclical pattern every four exponents:
[tex]\[
\begin{aligned}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1, \\
i^5 & = i, \\
\end{aligned}
\][/tex]
and this pattern repeats every four exponents.
2. Determine the Remainder When Dividing by 4:
Since the powers of [tex]\( i \)[/tex] cycle every four, we can determine the position within the cycle of [tex]\( i^{79} \)[/tex] by calculating [tex]\( 79 \mod 4 \)[/tex]:
[tex]\[
79 \div 4 = 19 \text{ remainder } 3
\][/tex]
Thus,
[tex]\[
79 \equiv 3 \pmod{4}
\][/tex]
3. Find [tex]\( i^{79} \)[/tex] in the Cycle:
From the cyclic pattern, we see:
[tex]\[
\begin{aligned}
i^3 & = -i \\
\end{aligned}
\][/tex]
Since [tex]\( 79 \mod 4 = 3 \)[/tex], it follows that:
[tex]\[
i^{79} = i^3 = -i
\][/tex]
So, based on the cyclic nature of powers of [tex]\( i \)[/tex], the value of [tex]\( i^{79} \)[/tex] is [tex]\(-i\)[/tex].
The correct answer is [tex]\( -i \)[/tex]. Thus, the corresponding answer choice is:
[tex]\[ \boxed{2} \][/tex]
