/*! 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 66 Find the indicated root, or stat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 66

Find the indicated root, or state that the expression is not a real number. $$\sqrt[7]{-1}$$

Short Answer

Expert verified
The seventh root of -1 is -1.

Step by step solution

01

Understand the Problem

The expression we need to evaluate is the 7th root of -1, denoted as \(\sqrt[7]{-1}\). The question will be solved using the property of roots that allows for negative roots when the root is an odd number.
02

Applying Root Properties

As the root number is odd (7 in this case), it is possible to find the 7th root of -1 in the set of real numbers. Any real number, raised to an odd power, will maintain its sign. Therefore, \(\sqrt[7]{-1}\) simply equals to -1 since (-1) raised to the power of 7 still gives -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.

Real Numbers
Real numbers include both rational and irrational numbers. They form a continuous number line that encompasses all numbers we typically use in daily life.
  • Rational numbers: These can be expressed as fractions or ratios, like \( \frac{1}{2} \) or \( 3 \).
  • Irrational numbers: These cannot be written as simple fractions, like \( \pi \) or \( \sqrt{2} \).
Real numbers allow for operations such as addition, subtraction, multiplication, and division. When dealing with roots, we primarily consider whether these values remain within the realm of real numbers.
Odd Roots
Odd roots, like the 7th root, allow us to find roots of negative numbers. This is a special property compared to even roots.
  • Odd roots: Always result in a real number, even for negative inputs.
  • Even roots: For negative numbers do not yield a real number without involving imaginary numbers.
For example, finding the 7th root (\( \sqrt[7]{-1} \)) is possible because -1 raised to an odd power still results in -1. This preserves the number's original sign.
Negative Numbers
Negative numbers represent values less than zero and are essential in various mathematical operations, including roots.
  • Properties: When multiplied by themselves an even number of times, they become positive.
  • Odd number of times: They maintain their negative sign.
Because odd roots can handle negative input, finding roots like \( \sqrt[7]{-1} \) results in another negative, real number: -1.
Algebra
Algebra involves symbols and rules for manipulating those symbols to solve equations. With roots, we use algebra to simplify expressions and find values.
  • Simplifying: Algebra helps us express roots in simpler forms.
  • Solving: We use algebraic techniques to find the values of roots and verify their correctness.
In solving \( \sqrt[7]{-1} \), algebra aids in understanding why raising -1 to any odd power will still equal -1.

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 \(53-74\), rationalize each denominator. Simplify, if possible. $$ \frac{1}{4-\sqrt{x}} $$

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\begin{aligned} &\frac{\sqrt{3}}{\sqrt{3}+1}\\\ &- \end{aligned}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$16^{-\frac{1}{4}}=-2$$

Explain why \(a^{\bar{n}}\) is negative when \(n\) is odd and \(a\) is negative. What happens if \(n\) is even and \(a\) is negative? Why?

In Exercises \(75-82\), rationalize each denominator. Simplify, if possible $$\frac{\sqrt{100 x^{5} y^{2}}}{\sqrt{2 x y^{3}}}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.