Which of the following is equivalent to $$ i^{8} $$ ?

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UpStudy AI · Accepted Answer

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

$$
\boxed{1}.
$$
The equivalent of $$ i^8 $$ is 1.

Which of the following is equivalent to ?

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