To determine the value of [tex]\( i^{84} \)[/tex], we start by recalling the fundamental properties of the imaginary unit [tex]\( i \)[/tex]. The imaginary unit [tex]\( i \)[/tex] is defined such that [tex]\( i = \sqrt{-1} \)[/tex]. Key powers of [tex]\( i \)[/tex] exhibit a repeating pattern every four exponents:
[tex]\[
i^1 = i
\][/tex]
[tex]\[
i^2 = -1
\][/tex]
[tex]\[
i^3 = -i
\][/tex]
[tex]\[
i^4 = 1
\][/tex]
Given that the powers of [tex]\( i \)[/tex] repeat every four exponents, we can utilize this periodicity to determine [tex]\( i^{84} \)[/tex]. Specifically, we observe the cycle [tex]\( i^1, i^2, i^3, i^4 \)[/tex] repeats.
To find [tex]\( i^{84} \)[/tex], we can reduce the exponent by applying modulo 4, because every four exponents, the cycle starts anew:
[tex]\[
84 \mod 4 = 0
\][/tex]
This tells us that 84 is exactly divisible by 4, meaning [tex]\( i^{84} \)[/tex] falls at the same position in the cycle as [tex]\( i^0 \)[/tex] (which would be another way to express [tex]\( i^4 \)[/tex]).
From the repeating pattern, we observe:
[tex]\[
i^4 = 1
\][/tex]
Therefore, since [tex]\( 84 \equiv 0 \mod 4 \)[/tex], we find that:
[tex]\[
i^{84} = i^0 = (i^4)^{21} = 1^{21} = 1.
\][/tex]
Thus, the value of [tex]\( i^{84} \)[/tex] is [tex]\( 1 \)[/tex].
Hence, the correct answer is [tex]\( \boxed{1} \)[/tex].
