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

Understand Powers of i: To find the equivalent of i^(32), we need to know the pattern of powers of i. The powers of i repeat in a cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. After i^4, the cycle repeats.Simplify i^(32): Since the powers of i repeat every 4 steps, we can simplify i^(32) by dividing 32 by 4. The remainder of this division will tell us the equivalent power of i within the first cycle of 4.Calculate i^(4*8): Divide 32 by 4: 32 ÷ 4 = 8. Since there is no remainder, i^(32) is equivalent to i^(4*8), which is (i^4)^8.Find Equivalent of i^4: We know that i^4 = 1. Therefore, (i^4)^8 = 1^8.Calculate 1^8: Calculate 1^8: 1^8 = 1, because any non-zero number raised to any power is itself.Conclude i^(32): Since 1^8 = 1, we conclude that i^(32) is equivalent to 1.

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Math Problems

Algebra 2

Quadratic equation with complex roots

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

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i^{32}

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Choose

1

answer:

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(A)

1

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(B)

i

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(C)

-1

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(D)

-i

Understand Powers of i: To find the equivalent of $i^{32}$ , we need to know the pattern of powers of $i$ . The powers of $i$ repeat in a cycle of $4$ : $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , and $i^4 = 1$ . After $i^4$ , the cycle repeats.
Simplify $i^{32}$ : Since the powers of $i$ repeat every $4$ steps, we can simplify $i^{32}$ by dividing $32$ by $4$ . The remainder of this division will tell us the equivalent power of $i$ within the first cycle of $4$ .
Calculate $i^{4\times8}$ : Divide $32$ by $4$ : $32 \div 4 = 8$ . Since there is no remainder, $i^{32}$ is equivalent to $i^{4\times8}$ , which is $(i^4)^8$ .
Find Equivalent of $i^4$ : We know that $i^4 = 1$ . Therefore, $(i^4)^8 = 1^8$ .
Calculate $1^8$ : Calculate $1^8$ : $1^8 = 1$ , because any non-zero number raised to any power is itself.
Conclude $i^{32}$ : Since $1^8 = 1$ , we conclude that $i^{32}$ is equivalent to $1$ .