Evaluate the power of $i$.
$i^{23}$
The complex number \(i\) is defined by the property \(i^2 = -1\). Powers of \(i\) follow a predictable cyclic pattern that repeats every four steps. Specifically, we have:
\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = -i\)
\(i^4 = 1\)
This pattern repeats because \(i^5 = i^4 \cdot i = 1 \cdot i = i\), and so on. Any power \(i^n\) can be calculated by finding the remainder when the exponent \(n\) is divided by 4, as \(i^n = i^r\) where \(r\) is the remainder of \(n\) divided by 4.
To evaluate \(i^{23}\), we divide the exponent 23 by 4:
\(23 = 4 \times 5 + 3\)
This division shows that the quotient is 5 and the remainder is 3. Using the laws of exponents, we can rewrite the expression as:
\(i^{23} = i^{4 \times 5 + 3} = (i^4)^5 \cdot i^3\)
Since \(i^4 = 1\), the expression simplifies to:
\(i^{23} = (1)^5 \cdot i^3 = 1 \cdot i^3 = i^3\)
We know from our analysis of the powers of \(i\) that \(i^3 = -i\). Therefore, the value of \(i^{23}\) is \(-i\). Comparing this to the given options:
◯ A. \(i\)
◯ B. \(-i\)
◯ C. \(-1\)
◯ D. \(1\)
The correct option is B.
Understanding the cycles of the imaginary unit is essential in fields like electrical engineering and signal processing. For instance, when analyzing alternating current (AC) circuits, engineers use complex numbers to represent impedance and phase shifts. The periodicity of these components allows them to simplify complex oscillations and waveforms, much like how we reduced the high exponent in this problem to a simple power of \(i\).