/*! 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 109 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 109

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My work with complex numbers verified that the only possible cube root of 8 is 2

Short Answer

Expert verified
The statement does not make sense. While 2 is indeed a cube root of 8 in the real number system, there are other solutions in the complex number system.

Step by step solution

01

Understanding Cube Roots

In the real number system, a cube root of a number \(n\) is a value \(a\) such that \(a^3 = n\). So, in the real number system, the cube root of 8 is indeed 2.
02

Dealing with Complex Numbers

Understand that in the complex number system, a number can actually have more than one cube root. This is because complex numbers involve both real and imaginary components, and 鈥渃ubing鈥� a complex number involves rotating it in the complex plane. The cube roots of a number are evenly distributed around a circle in the complex plane and are 120 degrees apart.
03

Conclusion and Reasoning

In the context of complex numbers, the statement that 'the only possible cube root of 8 is 2' is incorrect. In the complex number system, 8 actually has three cube roots. One of these is the real number 2, and the other two are complex numbers: \(-1 + \sqrt{3}i\) and \(-1 - \sqrt{3}i\). Therefore, the statement in the problem does not make sense, because it inaccurately claims that there is only one cube root of 8.

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!

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

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex cube roots of \(i\)

You want to fly your small plane due north, but there is a 75-kilometer wind blowing from west to east. a. Find the direction angle for where you should head the plane if your speed relative to the ground is 310 kilometers per hour. b. If you increase your airspeed, should the direction angle in part (a) increase or decrease? Explain your answer.

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$\begin{aligned}&r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15} \quad \text { (Use } \quad \theta \min =0 \quad \text { and }\\\&\theta \max =20 \pi .)\end{aligned}$$

Exercises \(119-121\) will help you prepare for the material covered in the next section. Find the obtuse angle \(\theta,\) rounded to the nearest tenth of a degree, satisfying. $$ \cos \theta=\frac{3(-1)+(-2)(4)}{| \mathbf{v}\|\mathbf{w}\|} $$ where \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{w}=-\mathbf{i}+4 \mathbf{j}\)

Verify the identity: $$\frac{1+\sin x}{1-\sin x}-\frac{1-\sin x}{1+\sin x}=4 \tan x \sec x$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

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.