What is the value of [tex]$i^{20+1}$[/tex]?
A. 1
B. -1
C. -1
D. [tex]$i$[/tex]
Answer (2)
Simplify the exponent: 20 + 1 = 21 , so we need to find i 21 .
Recognize the cyclic nature of i : i 1 = i , i 2 = − 1 , i 3 = − i , i 4 = 1 .
Divide the exponent by 4 and find the remainder: 21 ÷ 4 gives a remainder of 1.
Determine the value of i 21 using the remainder: i 21 = i 1 = i , so the final answer is i .
Explanation
Simplify the exponent We are asked to find the value of i 20 + 1 , where i is the imaginary unit. Let's break this down step by step. First, we need to simplify the exponent.
Calculate the exponent The exponent is 20 + 1 = 21 . So, we need to find the value of i 21 .
Understanding the cycle of i We know that the powers of i repeat in a cycle of 4:
i 1 = i i 2 = − 1 i 3 = − i i 4 = 1
To find i 21 , we can divide the exponent 21 by 4 and find the remainder. The remainder will tell us which value in the cycle corresponds to i 21 .
Find the remainder When we divide 21 by 4, we get 21 ÷ 4 = 5 with a remainder of 1. This means that i 21 is the same as i 1 .
Final Answer Since the remainder is 1, i 21 = i 1 = i . Therefore, the value of i 20 + 1 is i .
Examples
Understanding imaginary numbers is crucial in electrical engineering, especially when analyzing AC circuits. The impedance of a circuit, which is the opposition to the flow of alternating current, is often expressed using complex numbers. For example, if a circuit has a resistor and an inductor, the impedance might be represented as Z = R + i X L , where R is the resistance, X L is the inductive reactance, and i is the imaginary unit. Calculations involving powers of i help engineers determine the phase relationships between voltage and current in these circuits, ensuring efficient and stable operation.
The value of i 20 + 1 simplifies to i 21 , which equals i because it corresponds to the first power in the repeating cycle of powers of i . Therefore, the correct answer is D . i .
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