Question
Which of the following is equivalent to \( i^{8} \) ?
Ask by Grant Knight. in the United States
Nov 25,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The equivalent of \( i^8 \) is 1.
Solution
To find \( i^8 \), we start by recalling the powers of the imaginary unit \( i \), where \( i = \sqrt{-1} \). The powers of \( i \) cycle every four terms:
\[
\begin{align*}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1.
\end{align*}
\]
After \( i^4 \), the pattern repeats:
\[
\begin{align*}
i^5 & = i, \\
i^6 & = -1, \\
i^7 & = -i, \\
i^8 & = 1.
\end{align*}
\]
To compute \( i^8 \), we can also use the fact that the powers of \( i \) repeat every four terms. We can find the equivalent power by calculating \( 8 \mod 4 \):
\[
8 \div 4 = 2 \quad \text{(remainder 0)}.
\]
Since the remainder is 0, we have:
\[
i^8 = i^0 = 1.
\]
Thus, the equivalent of \( i^8 \) is
\[
\boxed{1}.
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find \( i^{8} \), we can look at the powers of \( i \) and see that they follow a cyclical pattern: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). These powers repeat every four terms, so we can simplify \( i^{8} \) as follows: Since \( 8 \div 4 = 2 \) with a remainder of \( 0 \), we find that \( i^{8} = (i^4)^{2} = 1^{2} = 1 \). So, \( i^{8} \) is equivalent to \( 1 \).
