Which of the following is equivalent to the complex number i^(51) ? Choose 1 answer: (A) 1 (B) i (C) -1 (D) -i

Recognizing the Pattern: To solve for i^(51), we need to recognize the pattern of powers of i. The powers of i cycle in a pattern: i, -1, -i, 1, and then repeat. This cycle repeats every 4 powers.Determining the Position in the Cycle: To find the equivalent of i^(51), we can divide 51 by the cycle length, which is 4, to determine where in the cycle i^(51) falls. 51 ÷ 4 = 12 remainder 3. This means that i^(51) is the same as i^(3) because after every full cycle of 4, the powers of i start repeating.Finding the Value of i^3: Now we need to find the value of i^(3). We know that i^(2) = -1, so i^(3) = i^(2) * i = -1 * i = -i.Final Result: Therefore, i^(51) is equivalent to -i, which corresponds to choice (D).

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Which of the following is equivalent to the complex number
i^(51) ?
Choose 1 answer:
(A) 1
(B)
i
(C) -1
(D)
-i

Which of the following is equivalent to the complex number $i^{51}$ ? $\newline$ 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

Full solution

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

i^{51}

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

i

\newline

(C)

-1

\newline

(D)

-i

Recognizing the Pattern: To solve for $i^{51}$ , we need to recognize the pattern of powers of $i$ . The powers of $i$ cycle in a pattern: $i$ , $-1$ , $-i$ , $1$ , and then repeat. This cycle repeats every $4$ powers.
Determining the Position in the Cycle: To find the equivalent of $i^{51}$ , we can divide $51$ by the cycle length, which is $4$ , to determine where in the cycle $i^{51}$ falls. $\newline$ $51 \div 4 = 12$ remainder $3$ . $\newline$ This means that $i^{51}$ is the same as $i^{3}$ because after every full cycle of $4$ , the powers of $i$ start repeating.
Finding the Value of $i^$ $3$ : Now we need to find the value of $i^{$ $3$ }. We know that $i^{$ $2$ } = $-1$ , so $i^{$ $3$ } = i^{ $2$ } \cdot i = $-1$ \cdot i = -i.
Final Result: Therefore, $i^{51}$ is equivalent to $-i$ , which corresponds to choice $(D)$ .