Write each power of i as i, - 1, - i, or 1. i44

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 55a

2장, 문제 55a

Write each power of i as i, - 1, - i, or 1.
i⁴⁴

검증된 단계별 안내

Step 1: Recall the powers of i follow a repeating cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers of i.

Step 2: To determine the value of i^44, divide the exponent (44) by 4 to find the remainder. This will help identify where in the cycle the power falls.

Step 3: Perform the division: 44 ÷ 4 = 11 with a remainder of 0. The remainder determines the position in the cycle.

Step 4: Since the remainder is 0, i^44 corresponds to i^4 in the cycle. From the cycle, we know that i^4 = 1.

Step 5: Conclude that i^44 simplifies to 1 based on the repeating cycle of powers of i.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Modulus Operation

To simplify powers of i, we can use the modulus operation with respect to 4, since the powers of i repeat every four terms. For example, to find i^44, we calculate 44 mod 4, which equals 0. This indicates that i^44 corresponds to i^0, which is equal to 1.

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 i, as it allows for the manipulation and interpretation of expressions involving imaginary units in various mathematical contexts.