/*! 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 63 Evaluate each expression in Exer... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 63

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[5]{(-3)^{5}}$$

Short Answer

Expert verified
-3

Step by step solution

01

Write the expression

The given expression is \(\sqrt[5]{(-3)^{5}}\).
02

Simplify the expression

Looking at the expression, it is evident that a number raised to a power, and then taking the root of that power is a reverse operation. The fifth root of a number raised to the power of 5 is the original number. In this case, the number is -3. So, \(\sqrt[5]{(-3)^{5}} = -3\).

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 are a central part of algebra and everyday mathematics. They combine both rational and irrational numbers. Rational numbers are fractions or integers like \(-2, 0,\) and \(0.75\). Irrational numbers include numbers that cannot be expressed as simple fractions, such as \(\sqrt{2}\) or \(\pi\). Together, these numbers form the continuum of real numbers that you can visualize on a number line.
  • Real numbers can be positive, negative, or zero.
  • They include special subsets like whole numbers, integers, and decimal numbers.
  • The concept of real numbers helps us solve equations and model real-world scenarios effectively.
When dealing with expressions like \[\sqrt[5]{(-3)^{5}}\], understanding that\(( -3 )\) is a real number assists in simplifying the expression, as real numbers are stable under operations like roots and exponents.
Exponents
Exponents are a shorthand way of expressing repeated multiplication of the same number by itself. For example, \(3^5\) means multiplying 3 by itself five times: \(3 \times 3 \times 3 \times 3 \times 3\).
  • An exponent tells you how many times the base is multiplied.
  • Negative exponents represent reciprocal values, like \(3^{-2} = \frac{1}{3^2}\).
  • Zero exponents also have a special rule: any nonzero number raised to the power zero is 1, written as \(n^0 = 1\).
In the expression \[\sqrt[5]{(-3)^{5}}\], the exponent is 5. The process essentially reverses itself when you then apply the fifth root, explaining why the simplified result is \(-3\). This operation demonstrates how exponents and roots can be inverse operations.
Roots
Roots, particularly square roots and higher roots, are the opposite operation to raising a number to a power. The notation \(\sqrt[n]{x}\) signifies the \(n\)th root of \(x\), and it is essentially asking, "What number raised to the \(n\)th power gives \(x\)?"
  • Finding a square root is common for \(n = 2\), denoted simply as \(\sqrt{x}\).
  • Higher roots, such as cube roots or fifth roots, are used similarly.
  • In the real numbers, only even roots of negative numbers are not real, whereas odd roots can be negative.
In the specific case of \[\sqrt[5]{(-3)^{5}}\], the fifth root cancels out raising \(-3\) to the fifth power, indicating that the number under the radical is indeed real and equal to \(-3\). This showcases how roots can simplify expressions when combined with their respective powers.

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 intersection of the sets. $$\\{1,3,7\\} \cap(2,3,8)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.

This will help you prepare for the material covered in the next section. a. Find \(\sqrt{16} \cdot \sqrt{4}\) b. Find \(\sqrt{16 \cdot 4}\) c. Based on your answers to parts (a) and (b), what can you conclude?

$$\text { Factor completely.}$$ $$(x-y)^{4}-4(x-y)^{2}$$

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-9,-\frac{4}{5}, 0,0.25, \sqrt{3}, 9.2, \sqrt{100}\right\\}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.