Which of the following is equivalent to the complex number i^(32) ? Choose 1 answer: (A) 1 (B) i (c) -1 (D) -i

Definition of i: To solve for i^(32), we need to remember that i is the imaginary unit, which is defined by i^2 = -1. We can use the powers of i to simplify i^(32).Powers of i: The powers of i repeat in a cycle: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats every 4 powers. Since 32 is a multiple of 4 (32 = 4 * 8), i^(32) will be the same as i^(4*8).Simplifying i^(32): Using the cycle, we know that i^4 = 1. Therefore, (i^4)^8 = 1^8.Using the cycle: 1 raised to any power is still 1, so 1^8 = 1.Simplifying 1^8: Therefore, i^(32) is equivalent to 1.

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Which of the following is equivalent to the complex number
i^(32) ?
Choose 1 answer:
(A) 1
(B)
i
(c) -1
(D)
-i

Which of the following is equivalent to the complex number $i^{32}$ ? $\newline$ 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^{32}

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

i

\newline

(C)

-1

\newline

(D)

-i

Definition of i: To solve for $i^{32}$ , we need to remember that $i$ is the imaginary unit, which is defined by $i^2 = -1$ . We can use the powers of $i$ to simplify $i^{32}$ .
Powers of i: The powers of i repeat in a cycle: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , and $i^4 = 1$ . This cycle repeats every $4$ powers. Since $32$ is a multiple of $4$ ( $32 = 4 \times 8$ ), $i^{32}$ will be the same as $i^{4\times8}$ .
Simplifying $i^{32}$ : Using the cycle, we know that $i^4 = 1$ . Therefore, $(i^4)^8 = 1^8$ .
Using the cycle: $1$ raised to any power is still $1$ , so $1^8 = 1$ .
Simplifying $1^8$ : Therefore, $i^{32}$ is equivalent to $1$ .