Which of the following is equivalent to the complex number i^(19) ? Choose 1 answer: (A) 1 (B) i (C) -1 (D) -i

Definition of i: To solve for i^(19), 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^(19) by finding a pattern.Pattern of powers of i: The powers of i repeat in a cycle: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Then the cycle repeats because i^5 = i, i^6 = -1, and so on.Finding i^(19): To find i^(19), we can divide 19 by 4 to find how many full cycles of 4 there are and what the remainder is. The remainder will tell us the equivalent power of i that we need to find. 19 ÷ 4 = 4 remainder 3. This means i^(19) is equivalent to i^(4*4 + 3), which is the same as i^(4*4) * i^3.Simplifying i^(19): Since i^(4*4) is a full cycle repeated 4 times, it is equivalent to 1. So we only need to consider i^3. i^3 = -i.Final result: Therefore, i^(19) is equivalent to i^(4*4) * i^3, which simplifies to 1 * -i, or just -i.

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Which of the following is equivalent to the complex number
i^(19) ?
Choose 1 answer:
(A) 1
(B)
i
(C) -1
(D)
-i

Which of the following is equivalent to the complex number $i^{19}$ ? $\newline$ 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^{19}

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

i

\newline

(C)

-1

\newline

(D)

-i

Definition of i: To solve for $i^{19}$ , 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 $i^{19}$ by finding a pattern.
Pattern of powers of i: The powers of i repeat in a cycle: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , and $i^4 = 1$ . Then the cycle repeats because $i^5 = i$ , $i^6 = -1$ , and so on.
Finding $i^{19}$ : To find $i^{19}$ , we can divide $19$ by $4$ to find how many full cycles of $4$ there are and what the remainder is. The remainder will tell us the equivalent power of $i$ that we need to find. $\newline$ $19 \div 4 = 4$ remainder $3$ . This means $i^{19}$ is equivalent to $i^{4\cdot4 + 3}$ , which is the same as $i^{19}$ $0$ .
Simplifying $i^{19}$ : Since $i^{4\times4}$ is a full cycle repeated $4$ times, it is equivalent to $1$ . So we only need to consider $i^3$ . $\newline$ $i^3 = -i$ .
Final result: Therefore, $i^{19}$ is equivalent to $i^{4\cdot4} \cdot i^3$ , which simplifies to $1 \cdot -i$ , or just $-i$ .