/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 92 Simplify each expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 92

Simplify each expression. $$ i^{22} $$

Short Answer

Expert verified
The expression \(i^{22}\) simplifies to \(-1\).

Step by step solution

01

Understand Powers of i

The imaginary unit, represented as \(i\), has a cyclical pattern with its powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). These four results repeat every four exponents. Therefore, any power of \(i\) can be reduced based on its remainder after division by 4.
02

Find the Remainder

To simplify \(i^{22}\), we first need to find the remainder when 22 is divided by 4, since every four exponents \(i\)'s pattern repeats. Calculating, we find \(22 \div 4 = 5\) with a remainder of 2. This means \(i^{22} = i^2\).
03

Apply the Remainder to i's Powers

Based on the powers of \(i\), we know that \(i^2 = -1\). Therefore, \(i^{22}\), which is equivalent to \(i^2\), simplifies to \(-1\).

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.

Understanding Powers of i
The concept of powers of the imaginary unit, denoted as \(i\), is fundamental when working with imaginary numbers. The value of \(i\) is defined as the square root of -1. However, the powers of \(i\) follow a predictable pattern, which simplifies calculations.

When \(i\) is raised to various exponents, the outcomes are simple and repeatable:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

This sequence repeats every four powers. This cyclical nature of \(i\)'s powers helps in simplifying expressions greatly. Any power of \(i\) can be mapped to one of these four outcomes, which is especially useful in more complex algebraic expressions involving imaginary numbers.
Recognizing Cyclical Patterns
Cyclical patterns in mathematics refer to sequences that repeat over and over in a regular interval. The powers of \(i\) exhibit this repetitive nature.

With \(i\), once you reach every fourth power, you circle back to the start of a pattern.
  • You have \(i^4 = 1\), \(i^5 = i\), and so on.
  • This cyclical behavior allows you to reduce the problem by making use of the remainder with respect to 4.

Essentially, to simplify any power of \(i\), divide the exponent by 4 and focus on the remainder. The remainder dictates which power from the initial cycle (\(i\), \(-1\), \(-i\), or \(1\)) corresponds to your expression.
Simplifying Expressions with Imaginary Numbers
When simplifying expressions involving imaginary numbers, we make use of the cyclical patterns of \(i\)'s powers. By doing so, we transform otherwise complex expressions into straightforward calculations.

Take the example \(i^{22}\). Instead of pondering 22 steps ahead, we divide 22 by 4, obtaining a remainder of 2. This leads us to the conclusion that \(i^{22} = i^2\).

Knowing the cyclical pattern, we quickly find \(i^2 = -1\). This response helps declutter expressions and fosters better comprehension in algebraic settings. Recognizing patterns and simplifying steps make understanding imaginary numbers less daunting for students.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Multiply and simplify. All variables represent positive real numbers. $$ (\sqrt{2}+1)(\sqrt{2}-3) $$

Fill in the blanks. An _____ triangle has three sides of equal length and three \(60^{\circ}\) angles.

Simplify each expression. All variables represent positive real numbers. $$ \left(16 a^{4}\right)^{-1 / 2} $$

Solve each equation. See Example 5. $$ (2 b+29)^{1 / 3}=3 $$

The number of mirrors that will be produced at a given price can be predicted by the formula \(s=\sqrt{23 x},\) where \(s\) is the supply (in thousands) and \(x\) is the price (in dollars). The demand \(d\) for mirrors can be predicted by the formula \(d=\sqrt{312-2 x^{2}} .\) Find the equilibrium price - that is, find the price at which supply will equal demand.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.