Problem 11 Calculate the given expression. ... [FREE SOLUTION]

Chapter 6: Problem 11

Calculate the given expression. \(i^{5}\)

Short Answer

Expert verified

The value of \(i^{5}\) is i.

Step by step solution

Understanding the properties of imaginary unit i

The imaginary unit, denoted as i, is defined by the property that \(i^2 = -1\). Therefore, expressions involving higher powers of i can be simplified using this definition.

Identify the pattern in powers of i

Notice that the powers of i repeat every four terms: \(i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1\). This cycle repeats, so \(i^5 = i^1 = i\).

Calculate i^5

Since the powers of i repeat every four terms, \(i^5\) can be determined by noticing that \(5 \text{ mod } 4 = 1\). Therefore, \(i^5 = i^1 = i\).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

powers of i

To understand the imaginary unit and its properties, let's delve into its powers. The symbol 'i' stands for the imaginary unit, defined by the equation: i² = -1. This special definition leads to a pattern in the powers of i. Observe the repeated sequence in its powers:

i¹ = i
i² = -1
i³ = -i
i⁴ = 1

This cyclical pattern continues indefinitely, meaning every fourth power of i will reset the sequence. Knowing this can greatly simplify calculations involving large powers of i.

i^2 = -1

A key property of the imaginary unit is that i² = -1. This fundamental definition helps manage the complexity of imaginary numbers. Here鈥檚 how it works:

When you square the imaginary unit i, you get -1: i²:

This property allows you to simplify higher powers of i by breaking them down using i².
For example, you can rewrite i³ as i² * i = -1 * i = -i.
Similarly, i⁴ simplifies to (i²)² = (-1)² = 1.

This property explains why the powers of i have such a predictable pattern.

modulus and cycle

To deal with large powers of i, understanding the concept of modulus in cycles is essential. Here鈥檚 a breakdown of how it works:

Recognize that the powers of i repeat every four terms. This periodicity means any power iⁿ can be reduced using modulo 4 arithmetic:

First, find the remainder when n is divided by 4 (denoted as n mod 4).
Then, use this remainder to redefine the power in the simplest form.

For example, with i⁵:

Compute 5 mod 4 = 1.
This tells us that i⁵ is equivalent to i¹.
Hence, i⁵ = i.

This method greatly simplifies working with higher powers of the imaginary unit.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Calculate the given expression. \(i^{5}\)

Short Answer

Step by step solution

Understanding the properties of imaginary unit i

Identify the pattern in powers of i

Calculate i^5

Key Concepts

powers of i

i^2 = -1

modulus and cycle

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Statistics

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.