Which expression is equivalent to $i^{233}$?
A. 1
B. -1
C. i
D. -i
Answer (2)
Divide the exponent 233 by 4 to find the remainder.
The remainder is 1.
Therefore, i 233 is equivalent to i 1 .
The expression equivalent to i 233 is i .
Explanation
Understanding the Problem We are asked to find an expression equivalent to i 233 , where i is the imaginary unit, defined as i = − 1 . We know that the powers of i cycle through four values: i 1 = i , i 2 = − 1 , i 3 = − i , and i 4 = 1 . To find i 233 , we need to determine where 233 falls in this cycle.
Finding the Remainder To find the equivalent expression, we divide the exponent 233 by 4 to find the remainder. This will tell us which value in the cycle i 233 corresponds to. The calculation is 233 ÷ 4 = 58 with a remainder of 1.
Determining the Equivalent Expression Since the remainder is 1, i 233 is equivalent to i 1 , which is simply i . Therefore, i 233 = i .
Final Answer The expression equivalent to i 233 is i .
Examples
Understanding the powers of i is crucial in electrical engineering when analyzing alternating current (AC) circuits. Impedance, which is the AC equivalent of resistance, is often expressed using complex numbers involving i . Calculating powers of i helps engineers simplify complex circuit equations and understand the behavior of AC circuits. For example, determining the impedance of a circuit component might involve simplifying expressions with i 233 to find the real and imaginary components of the impedance, which are essential for circuit design and analysis.
To find i 233 , we calculate the remainder of 233 divided by 4, which is 1. This tells us that i 233 is equivalent to i 1 , which is simply i . Therefore, the answer is i .
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