Which of the following is equivalent to the complex number i^(41) ? Choose 1 answer: (A) 1 (B) i (C) -1 (D) -i

Definition of i: To find the equivalent of i raised to any power, we need to remember that i is the imaginary unit, defined as i = √(-1). The powers of i repeat in a cycle: i, -1, -i, 1, and then back to i. This cycle repeats every 4 powers.Determining the Power Cycle: Let's find the remainder when 41 is divided by 4 to determine where in the cycle i^(41) will land. 41 divided by 4 is 10 with a remainder of 1.Finding the Remainder: Since the remainder is 1, i^(41) is equivalent to i^(4*10 + 1), which is the same as (i^4)^10 * i^1. We know that i^4 = 1, so this simplifies to 1^10 * i, which is just i.Simplifying the Expression: Therefore, i^(41) is equivalent to i. This corresponds to choice (B) in the given options.

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Which of the following is equivalent to the complex number
i^(41) ?
Choose 1 answer:
(A) 1
(B)
i
(C) -1
(D)
-i

Which of the following is equivalent to the complex number $i^{41}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $1$ $\newline$ (B) $i$ $\newline$ (C) $-1$ $\newline$ (D) $-i$

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Math Problems

Algebra 1

Domain and range of quadratic functions: equations

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Q. Which of the following is equivalent to the complex number

i^{41}

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

i

\newline

(C)

-1

\newline

(D)

-i

Definition of i: To find the equivalent of i raised to any power, we need to remember that i is the imaginary unit, defined as $i = \sqrt{-1}$ . The powers of i repeat in a cycle: $i$ , $-1$ , $-i$ , $1$ , and then back to $i$ . This cycle repeats every $4$ powers.
Determining the Power Cycle: Let's find the remainder when $41$ is divided by $4$ to determine where in the cycle $i^{41}$ will land. $\newline$ $41$ divided by $4$ is $10$ with a remainder of $1$ .
Finding the Remainder: Since the remainder is $1$ , $i^{41}$ is equivalent to $i^{4\cdot10 + 1}$ , which is the same as $(i^4)^{10} \cdot i^1$ . We know that $i^4 = 1$ , so this simplifies to $1^{10} \cdot i$ , which is just $i$ .
Simplifying the Expression: Therefore, $i^{41}$ is equivalent to $i$ . This corresponds to choice $(B)$ in the given options.