Powers of i Calculate the first 12 powers of i that is. i, i^2, i^3, …, i^12 Do you notice a pat

Name: Problem 82, Chapter 1: Algebra and Trigonometry
Uploaded: 2023-09-18T15:54:00-08:00
Duration: 2 min 22 s
Channel: Massimo Antonelli
Description: Powers of i Calculate the first 12 powers of i that is. i, i^2, i^3, …, i^12 Do you notice a pattern? Explain how you would calculate any whole number power of i, using the pattern that you have discovered. Use this procedure to calculate i^4446

step 1 All right, let's take a look at the powers of i and see how it's cyclical in nature going throug

Question

Powers of $i$ Calculate the first 12 powers of $i$ that is. $i, i^{2}, i^{3}, \ldots, i^{12}$ Do you notice a pattern? Explain how you would calculate any whole number power of $i,$ using the pattern that you have discovered. Use this procedure to calculate $i^{4446}$

   Powers of $i$ Calculate the first 12 powers of $i$ that is. $i, i^{2}, i^{3}, \ldots, i^{12}$ Do you notice a pattern? Explain how you would calculate any whole number power of $i,$ using the pattern that you have discovered. Use this procedure to calculate $i^{4446}$

Algebra and Trigonometry

James Stewart,… 3rd Edition

Chapter 1, Problem 82

Instant Answer

Solved by Expert Massimo Antonelli

09/18/2023

Step 1

We know that $i$ is the imaginary unit with the property that $i^{2} = -1$. Show more…

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Powers of $i$ Calculate the first 12 powers of $i$ that is. $i, i^{2}, i^{3}, \ldots, i^{12}$ Do you notice a pattern? Explain how you would calculate any whole number power of $i,$ using the pattern that you have discovered. Use this procedure to calculate $i^{4446}$

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Key Concepts

Imaginary Unit

The imaginary unit, typically represented as i, is defined by the property that i² = -1. It serves as the foundation for complex numbers, allowing the extension of the real number system to include solutions to equations like x² + 1 = 0.

Cyclic Pattern of Powers

When raised to consecutive integers, the powers of i follow a repeating cyclic pattern of four values: i, -1, -i, and 1. After every four powers, this cycle repeats itself. Recognizing this pattern simplifies calculations involving large exponents of i.

Modular Arithmetic

Modular arithmetic is used to determine positions within a repeating cycle. By computing the exponent modulo the cycle length (which is 4 for powers of i), one can efficiently determine the corresponding value in the cycle. This method is especially useful for calculating large exponents, such as finding the value of i raised to a very high power.

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