Example of Simplifying $i^{86}$

An engineering student is simplifying the expression $i^{47}$ to solve a problem involving alternating current phase shifts. Arrange the following steps in the correct order to simplify this power of the imaginary unit according to the standard mathematical procedure.

In technical fields such as telecommunications and electrical engineering, complex numbers are frequently used to model alternating current and signal phases. A core skill is simplifying powers of the imaginary unit $i$ by identifying the remainder when the exponent is divided by 4. Match each possible remainder to its simplified equivalent value.

In a technical training manual for electronics, a technician is taught to simplify powers of the imaginary unit $i$ by dividing the exponent by ${}4$. According to the fundamental properties of complex numbers, what is the specific value of $i^4$ that justifies this simplification method?

In a technical training course for electronics technicians, students are taught to simplify high powers of the imaginary unit $i$ for use in circuit analysis. The rule states that for any positive integer exponent $n$, the expression $i^n$ simplifies to $i^r$, where $r$ represents the ____ obtained when the exponent $n$ is divided by 4.

In an electrical engineering bridging course, adult learners are taught that to simplify a high power of the imaginary unit $i$, such as $i^n$, they must divide the exponent $n$ by ${}4$ and use the remainder to determine the final equivalent value.