/*! 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 4 For Problems \(1-30\), evaluate ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

For Problems \(1-30\), evaluate each numerical expression. $$ (-32)^{\frac{1}{5}} $$

Short Answer

Expert verified
\((-2)\) satisfies \((-32)^{\frac{1}{5}}\).

Step by step solution

01

Understand the Expression

The expression \((-32)^{\frac{1}{5}}\) involves finding the fifth root of \(-32\). This means we need to find a number that, when raised to the power of \(5\), equals \(-32\).
02

Evaluate the Fifth Root Conceptually

Since the expression involves an odd root, the result can be negative. We need to determine what number multiplied by itself four more times equals \(-32\).
03

Identify the Number

Start by considering smaller numbers: \((-2)^5 = (-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\). Thus, \((-2)\) is the number that satisfies the root condition.
04

Verify Your Result

Check your result by confirming that \((-2)^5 = -32\), ensuring the calculation correctly evaluates to the given expression.

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 Numerical Expressions
A numerical expression is a mathematical phrase that includes numbers, operations, and sometimes variables. It does not have an equality or inequality sign.
For example, in the given expression \((-32)^{\frac{1}{5}}\), you have a base of \(-32\) and an exponent of \(\frac{1}{5}\). This expression requires evaluating the fifth root of \(-32\).
In any numerical expression involving exponents, numbers are raised to a power. Here, the number \(-32\) is raised to the power of \(\frac{1}{5}\) indicating the fifth root. We express such roots through fractional exponents where the denominator represents the root involved. Particularly, \(\frac{1}{5}\) denotes the fifth root.
Negative Numbers and Exponentiation
When dealing with exponentiation of negative numbers, it's crucial to consider whether the exponent is an integer or a fractional power.
  • If the exponent is an integer, multiplying the number by itself involves repeating the process for as many times as the exponent indicates. Powers with even integers result in positive values because multiplying a negative number by itself an even number of times yields a positive number.
  • If the exponent is a fractional power, it turns into a root operation. This changes the approach because fractions represent roots, like in our expression \((-32)^{\frac{1}{5}}\).
It's important to note that fractional exponents allow us to compute roots, and they can cope with negative bases provided the root's order is odd. This concept is essential in understanding how the fifth root can exist for a negative base number.
Odd Roots and Negative Bases
When evaluating odd roots of negative numbers, keep in mind that the result will also be negative. Let's illustrate this using our example \((-32)^{\frac{1}{5}}\).
The fifth root in this case asks "What number multiplied by itself five times equals -32?". By trying \(-2\), you can see:
  • \((-2) \times (-2) = 4\)
  • \(4 \times (-2) = -8\)
  • \(-8 \times (-2) = 16\)
  • \(16 \times (-2) = -32\)
Therefore, \(-2\) is indeed the fifth root of \(-32\). The key takeaway is that odd roots of negatives remain negatives, allowing us to compute values for expressions with odd root orders.

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

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt[3]{16}}{\sqrt[6]{4}} $$

For Problems \(1-18\), write each of the following in scientific notation. \(0.00000000194\)

Explain how you would evaluate \(27^{\frac{2}{3}}\) without a calculator.

Perform the indicated operations and express answers in simplest radical form. $$ \frac{\sqrt{2}}{\sqrt[3]{2}} $$

A square pixel on a computer screen has a side of length (1.17) \(\left(10^{-2}\right)\) inches. Find the approximate area of the pixel in inches. Express the result in decimal form.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.