/*! 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 62 Simplify. Use absolute-value not... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 62

Simplify. Use absolute-value notation when necessary. $$ \sqrt[5]{-\frac{1}{32}} $$

Short Answer

Expert verified
-\( \frac{1}{2} \)

Step by step solution

01

Rewrite the expression

Rewrite the given expression \(\root[5]{-\frac{1}{32}}\) in a more workable form. Notice that \(-\frac{1}{32}\) can be written as \(\frac{-1}{32}\) and that \(\frac{1}{32} = \frac{1}{2^5} = 2^{-5}\). Therefore, \(-\frac{1}{32} = -2^{-5} \)
02

Distribute the radical exponent

Since we are dealing with the fifth root, apply the exponent \( \frac{1}{5} \) to the fraction inside the radical. This gives: \(\root[5]{-2^{-5}} = (-2^{-5})^{\frac{1}{5}}\)
03

Simplify the exponent

Simplify the exponent by multiplying \(-2^{-5}\) by \( \frac{1}{5} \): \((-2^{-5})^{\frac{1}{5}} = -2^{-1}\)
04

Evaluate the simplified expression

Evaluating \(-2^{-1}\) gives \(- \frac{1}{2}\). Hence, the simplified form is \(-\frac{1}{2}\)

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.

Radical Expressions
Radical expressions involve roots, such as square roots or fifth roots. The number under the radical sign is called the 'radicand'. For example, in \(\root[5]{-\frac{1}{32}}\), the radicand is \(-\frac{1}{32}\). To simplify, express the radicand in a more manageable form. Here, \(-\frac{1}{32}\) can be written as \(\frac{-1}{32}\). Knowing the properties of exponents helps in breaking down radical expressions to simpler forms.

Key steps include: expressing the radicand in terms of exponents, and distributing the radical to each part of the fraction within the root.
Exponents
Exponents indicate how many times a number (the base) is multiplied by itself. In this exercise, we use exponents to simplify the expression under the radical. Recognize that \(\frac{1}{32}\) can be rewritten using exponents as \(\frac{1}{2^5} = 2^{-5}\).

When working with negative exponents:
  • A negative exponent means taking the reciprocal of the base. For example, \(2^{-5} = \frac{1}{2^5}\).
  • Multiplying exponents: When raising a power to another power, multiply the exponents together. For instance, \((-2^{-5})^{\frac{1}{5}} = -2^{-1}\).
  • The power of a product: To simplify \((-2^{-5})^{\frac{1}{5}}\), apply the fractional exponent to both the base (2) and exponent (-5).
Fifth Roots
Fifth roots involve distributing the radical exponent (1/5) to the base inside the radical. In our example, we simplified \(\root[5]{-\frac{1}{32}}\). Here are the key steps:

  • Rewrite the radicand with exponents: \(\frac{1}{32} = 2^{-5}\).
  • Combine the radical exponent: Applying the fifth root to \(-2^{-5}\) means multiplying the existing exponent by 1/5: \((-2^{-5})^{\frac{1}{5}} = -2^{-1}\). This simplifies to \(-\frac{1}{2}\).

By understanding and applying these steps in a structured manner, students can master simplifying radical expressions involving fifth roots.

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 midpoint of the segment with the given endpoints. $$ (1,4) \text { and }(9,-6) $$

Multiply. $$ \sqrt{9+3 \sqrt{5}} \sqrt{9-3 \sqrt{5}} $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \sqrt{2 x^{3} y^{3}} \sqrt[3]{4 x y^{2}} $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non-negative real numbers. $$ f(x)=\sqrt[4]{2 x}+5 \sqrt{2 x}, g(x)=\sqrt[3]{2 x} $$

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|-1+i|$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.