Which value is equivalent to $i^{57}$?
The problem asks us to evaluate the expression \(i^{57}\), where \(i\) is the imaginary unit defined by the property \(i^2 = -1\). To solve this, we must identify the pattern of the powers of the imaginary unit \(i^n\). These powers follow a periodic cycle of four:
\[i^1 = i\]
\[i^2 = -1\]
\[i^3 = i^2 \cdot i = -1 \cdot i = -i\]
\[i^4 = (i^2)^2 = (-1)^2 = 1\]
Since \(i^4 = 1\), any higher power of \(i\) can be simplified by dividing the exponent by 4 and observing the remainder. Specifically, for any positive integer \(n\), \(i^n = i^{4k+r} = (i^4)^k \cdot i^r = 1^k \cdot i^r = i^r\), where \(r\) is the remainder when \(n\) is divided by 4.
To simplify \(i^{57}\), we divide the exponent 57 by 4 to find the remainder:
\[57 = 4 \times 14 + 1\]
This calculation shows that the quotient is 14 and the remainder is 1. Thus, we can express the exponent as:
\[57 \equiv 1 \pmod{4}\]
Using the property of the cycle of 4, we have:
\[i^{57} = i^{4 \times 14 + 1} = (i^4)^{14} \cdot i^1\]
Since \(i^4 = 1\), this simplifies to:
\[1^{14} \cdot i = 1 \cdot i = i\]
Having simplified \(i^{57}\) to \(i\), we check the given options:
◯ A. \(1\)
◯ B. \(-i\)
◯ C. \(-1\)
◯ D. \(i\)
The calculated value \(i\) corresponds directly to option D. Therefore, the equivalent value is \(i\). \(\boxed{i}\)
Understanding the cycle of powers of the imaginary unit is essential in fields like electrical engineering, where complex numbers are used to analyze alternating current (AC) circuits. Calculating powers of \(i\) helps in simplifying complex impedance equations, allowing engineers to determine phase shifts and magnitude changes in circuits more efficiently. This concept is a fundamental tool for solving problems involving periodic oscillation and wave phenomena.