/*! 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 In Exercises \(39-64,\) rational... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 62

In Exercises \(39-64,\) rationalize each denominator. $$\frac{10}{\sqrt[5]{16 x^{2}}}$$

Short Answer

Expert verified
The rationalized denominator is \(\frac{100000}{16x^{2}}\).

Step by step solution

01

Identify the denominator

The denominator of the given expression is \(\sqrt[5]{16 x^{2}}\). So our goal is to eliminate this 5th root from the denominator.
02

Rationalize the denominator

To eliminate the 5th root, raise both the numerator and denominator to the power of 5. So, \(\left(\frac{10}{\sqrt[5]{16 x^{2}}}\right)^{5} = \frac{10^{5}}{(16x^{2})}\).
03

Simplify the expression

Now, just simplify the expression. \(10^{5} = 100000\) and \(16x^{2}\) remains the same. So the final expression becomes \(\frac{100000}{16x^{2}}\).

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.

Simplifying Expressions
In algebra, simplifying expressions is crucial for making complex problems easier to solve and understand. The goal is to rewrite the expression in its simplest form without changing its value.
  • This involves combining like terms, which are terms that have the same variables raised to the same powers.
  • Another aspect is reducing fractions by dividing both the numerator and denominator by their greatest common factor.

In our exercise, simplifying refers to both rationalizing the denominator and performing arithmetic simplifications, such as when we find that \(10^5 = 100000\). This step is essential to represent the expression in its simplest form, making it easier to interpret and use in further mathematical operations.
Fifth Roots
Fifth roots, represented as \(\sqrt[5]{a}\), are values that, when raised to the power of 5, equal the given number \(a\). Simplifying expressions with fifth roots often involves eliminating these roots for a more straightforward number representation.
  • Fifth roots serve as an inverse operation to raising a number to the power of five. For instance, finding the fifth root of 32 gives 2, because \(2^5 = 32\).
  • In algebra, handling fifth roots typically involves multiplying by appropriate expressions to rationalize denominators. This makes calculations simpler and results clearer.

In the specific exercise, rationalizing the denominator means removing the fifth root by raising both parts of the fraction to the necessary power to clear the root, leading to a simplified expression.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Simplifying algebraic fractions involves several steps to make them more manageable.
  • The key steps include factoring polynomials, canceling common factors from the numerator and the denominator, and rationalizing denominators.
  • Rationalizing involves eliminating roots or radicals from the denominator by multiplying by the appropriate form of 1 (like \(\frac{\sqrt[5]{16x^3}}{\sqrt[5]{16x^3}}\) in this case).

For instance, in the expression \(\frac{10}{\sqrt[5]{16x^2}}\), by rationalizing the denominator, the expression becomes simpler and more logical to work with in more complex algebraic problems.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equations \(\sqrt{x+4}=-5\) and \(x+4=25\) have the same solution set.

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{12}{\sqrt{7}+\sqrt{3}}$$

In Exercises \(93-104\), rationalize each numerator. Simplify, if possible. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

Explain how to perform this multiplication: \((2+\sqrt{3})^{2}\)

Determine whether each relation is a function. (Section 8.1, Example 2) a. \(\\{(-1,1),(1,1),(-2,4),(2,4)\\}\) b. \([(1,-1),(1,1),(4,-2),(4,2)]\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.