To find the value of [tex]\( i^7 \)[/tex], we need to understand the properties of powers of the imaginary unit [tex]\( i \)[/tex]. The imaginary unit [tex]\( i \)[/tex] has a cyclical pattern in its powers, repeating every four terms. This cyclical pattern is as follows:
[tex]\[
\begin{align*}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1, \\
i^5 & = i, \\
i^6 & = -1, \\
i^7 & = -i, \\
i^8 & = 1, \quad \text{and so on.}
\end{align*}
\][/tex]
Notice that every four powers, the pattern repeats itself: [tex]\( i, -1, -i, 1 \)[/tex].
To determine [tex]\( i^7 \)[/tex], we can first simplify the exponent 7 in terms of this 4-term cycle. We do this by finding the remainder when 7 is divided by 4:
[tex]\[
7 \div 4 = 1 \text{ with a remainder of } 3.
\][/tex]
This remainder tells us the equivalent power within one full cycle. Specifically, the remainder 3 indicates that [tex]\( i^7 \)[/tex] is equivalent to [tex]\( i^3 \)[/tex].
From the established cycle, we know that:
[tex]\[
i^3 = -i.
\][/tex]
Therefore, the value of [tex]\( i^7 \)[/tex] is:
[tex]\[
\boxed{-i}.
\][/tex]
So the correct answer is:
(C) [tex]\( -i \)[/tex].
