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mode eview later 1. Prove algebraically that i =1. A

mode eview later 1. Prove algebraically that i =1. A

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem Statement:** 4. Prove algebraically that \(i^4 = 1\). **Solution:** To prove that \(i^4 = 1\), we need to understand the powers of the imaginary unit \(i\), where \(i\) is defined as the square root of -1, i.e., \(i = \sqrt{-1}\). Let's examine the first few powers of \(i\): 1. \(i^1 = i\) 2. \(i^2 = i \times i = -1\) (by definition since \(i = \sqrt{-1}\)) 3. \(i^3 = i^2 \times i = -1 \times i = -i\) 4. \(i^4 = i^3 \times i = -i \times i = -i^2 = -(-1) = 1\) This completes the proof. The sequence of powers \(i^1, i^2, i^3, i^4\) repeats every 4 steps. Therefore, \(i^4 = 1\), and this pattern will hold for every integer multiple of 4. **Conclusion:** Thus, we have proven algebraically that \(i^4 = 1\).
Transcribed Image Text:**Problem Statement:** 4. Prove algebraically that \(i^4 = 1\). **Solution:** To prove that \(i^4 = 1\), we need to understand the powers of the imaginary unit \(i\), where \(i\) is defined as the square root of -1, i.e., \(i = \sqrt{-1}\). Let's examine the first few powers of \(i\): 1. \(i^1 = i\) 2. \(i^2 = i \times i = -1\) (by definition since \(i = \sqrt{-1}\)) 3. \(i^3 = i^2 \times i = -1 \times i = -i\) 4. \(i^4 = i^3 \times i = -i \times i = -i^2 = -(-1) = 1\) This completes the proof. The sequence of powers \(i^1, i^2, i^3, i^4\) repeats every 4 steps. Therefore, \(i^4 = 1\), and this pattern will hold for every integer multiple of 4. **Conclusion:** Thus, we have proven algebraically that \(i^4 = 1\).
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