Write each power of i as i, - 1, - i, or 1. i135

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 60

2장, 문제 60

Write each power of i as i, - 1, - i, or 1. i¹³⁵

검증된 단계별 안내

Understand that the powers of i (the imaginary unit) follow a repeating cycle: i, -1, -i, 1. This cycle repeats every 4 powers.

To determine the value of i^135, divide the exponent 135 by 4 and find the remainder. This is because the cycle repeats every 4 powers.

Perform the division: 135 ÷ 4. The quotient is 33, and the remainder is 3. The remainder determines the position in the cycle.

Match the remainder to the cycle: A remainder of 1 corresponds to i, 2 corresponds to -1, 3 corresponds to -i, and 0 corresponds to 1.

Since the remainder is 3, i^135 corresponds to the third position in the cycle, which is -i.

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주요 개념

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Powers of i

The imaginary unit i is defined as the square root of -1. Its powers cycle through four distinct values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cyclical pattern repeats every four powers, which is crucial for simplifying higher powers of i.

Modulo Operation

To simplify powers of i, we can use the modulo operation. Specifically, we find the exponent modulo 4, since the powers of i repeat every four terms. For example, to simplify i^135, we calculate 135 mod 4, which helps determine the equivalent lower power of i.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. Understanding complex numbers is essential for working with powers of i, as they form the basis of operations involving imaginary units.