Write each power of i as i, - 1, - i, or 1.i114

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 57a

2장, 문제 57a

Write each power of i as i, - 1, - i, or 1.
i¹¹⁴

검증된 단계별 안내

Step 1: Recall the cyclical nature 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. This cycle repeats for higher powers.

Step 2: To determine the value of i^114, divide the exponent (114) by 4 and find the remainder. This is because the cycle repeats every 4 powers.

Step 3: Perform the division 114 ÷ 4. The quotient is 28, and the remainder is 2. This means i^114 is equivalent to i^2.

Step 4: Refer back to the cycle of powers of i. From the cycle, i^2 = -1.

Step 5: Conclude that i^114 simplifies to -1 based on the cyclical pattern of powers of i.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 and Division

To simplify powers of i, we can use the modulus of the exponent. Since the powers of i repeat every four terms, we can find the equivalent power by calculating the exponent modulo 4. For example, i^114 can be simplified by finding 114 mod 4, which helps determine the corresponding value in the cycle.

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.