Simplify each power of i. i11

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 31

2장, 문제 31

Simplify each power of i. i¹¹

검증된 단계별 안내

Recall that the imaginary unit \( i \) has the property \( i^2 = -1 \). Powers of \( i \) cycle every 4 steps: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then the pattern repeats.

To simplify \( i^{11} \), find the remainder when 11 is divided by 4, since the powers of \( i \) repeat every 4.

Calculate \( 11 \div 4 \) which gives a quotient of 2 and a remainder of 3, so \( 11 = 4 \times 2 + 3 \).

Rewrite \( i^{11} \) as \( i^{4 \times 2 + 3} = (i^4)^2 \times i^3 \).

Since \( i^4 = 1 \), simplify to \( 1^2 \times i^3 = i^3 \), and then use the cycle to express \( i^3 = -i \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Imaginary Unit i

The imaginary unit i is defined as the square root of -1, satisfying i² = -1. It is the fundamental building block for complex numbers and powers of i cycle through a pattern based on this definition.

Powers of i and Their Cyclic Pattern

Powers of i repeat every four exponents: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then the cycle repeats. This pattern helps simplify higher powers of i by reducing the exponent modulo 4.

Modular Arithmetic for Exponent Reduction

Modular arithmetic involves finding the remainder when dividing the exponent by 4 to simplify powers of i. For example, i¹¹ can be simplified by calculating 11 mod 4 = 3, so i¹¹ = i³ = -i.