/*! 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 9 Decide whether each expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Decide whether each expression is equal to \(1,-1, i,\) or \(-i .\) $$i^{2}$$

Short Answer

Expert verified
-1

Step by step solution

01

Understand the properties of the imaginary unit i

The imaginary unit \(i\) is defined such that \(i^2 = -1\).
02

Apply the property to the given expression

Given \(i^2\), we directly use the definition \(i^2 = -1\).

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.

Properties of Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in mathematics. It is defined by the property: \(i^2 = -1\). This means that when you square \(i\), you get \(-1\).
Imaginary numbers are used to extend the real number system \( \mathbb{R} \) to the complex number system \( \mathbb{C} \).
Here are some basic properties of the imaginary unit:
\(i^1 = i\)
\(i^2 = -1\)
\(i^3 = -i\)
\(i^4 = 1\)
Notice that powers of \(i\) repeat every four exponents. Recognizing these patterns can simplify many complex number calculations.
Complex Numbers
While real numbers are numbers we are most familiar with, complex numbers are used to solve equations involving square roots of negative numbers. A complex number can be written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
Here are some key points about complex numbers:
  • The real part of a complex number is \(a\).
  • The imaginary part of a complex number is \(b\).
  • Complex numbers can be added, subtracted, multiplied, and divided similar to real numbers.
Complex numbers have many applications in engineering, physics, and computer science.
i Squared
\(i^2\) is one of the key properties of the imaginary unit. As mentioned earlier, \(i^2 = -1\). This definition is what transforms the imaginary and real number systems into the complex number system.
[Recap Step-by-Step Solution]: To determine what \(i^2\) equals, we use its definition: \(i^2 = -1\). Therefore, recognizing that \(i^2 = -1\) is crucial in solving many problems involving complex numbers.

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

Find the distance between each pair of points. \((\sqrt{7}, 9 \sqrt{3})\) and \((-\sqrt{7}, 4 \sqrt{3})\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{\sqrt{8}}{3-\sqrt{2}} $$

The illumination \(I\), in foot-candles, produced by a light source is related to the distance \(d\), in feet, from the light source by the equation $$ d=\sqrt{\frac{k}{I}} $$ where \(k\) is a constant. If \(k=640,\) how far from the light source will the illumination be 2 foot-candles? Give the exact value, and then round to the nearest tenth of a foot.

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ -\sqrt{\frac{75 m^{3}}{p}} $$

Write each quotient in lowest terms. Assume that all variables represent positive real numbers. $$ \frac{6 p+\sqrt{24 p^{3}}}{3 p} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

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.