/*! 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 58 In Exercises \(39-64,\) rational... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 58

In Exercises \(39-64,\) rationalize each denominator. $$\sqrt[3]{\frac{3}{x y^{2}}}$$

Short Answer

Expert verified
The original expression \(\sqrt[3]{\frac{3}{x y^{2}}}\) when rationalized becomes \(\frac{3}{xy^2}\).

Step by step solution

01

Simplify the expression

Rewrite \(\sqrt[3]{\frac{3}{x y^{2}}}\) as \(\frac{\sqrt[3]{3}}{\sqrt[3]{xy^2}}\) by applying the rule of square roots division.
02

Rationalize the denominator

Cube both the numerator and denominator. By cubing \(\sqrt[3]{3}\), you obtain 3 and cube of \(\sqrt[3]{xy^2}\) is \(xy^2\). Therefore, the rationalized denominator expression becomes \(\frac{3}{xy^2}\).

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.

Cube Roots
A cube root is a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because \(3 \times 3 \times 3 = 27\). In mathematical terms, the cube root of a number \(a\) is represented as \(\sqrt[3]{a}\).
This is different from square roots, which only require multiplying by itself twice. Understanding cube roots is vital in solving algebraic problems, especially involving rational expressions.
  • Cube root of a product: \(\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}\)
  • The number \(a = b^3\) implies that \(b = \sqrt[3]{a}\)
Understanding how to simplify cube roots can make solving algebraic expressions easier, particularly when you need to rationalize denominators.
Simplifying Expressions
Simplifying an expression involves rewriting it in a more manageable form. This often means making the expression shorter or easier to understand and solve. For cube roots in algebraic expressions, simplification can mean rewriting fractions in a way that makes them easier to work with.
Take \(\sqrt[3]{\frac{3}{xy^2}}\), for example. Simplifying this expression involves separating the cube root over the numerator and the denominator, turning it into \(\frac{\sqrt[3]{3}}{\sqrt[3]{xy^2}}\).
  • Simplifying can also involve factorization and combining like terms.
  • This provides a "cleaner" version of the expression, which is especially useful when solving equations.
Simplifying expressions is a crucial part of many mathematical processes, including rationalizing denominators, which is required to remove any roots from the bottom of a fraction.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. They do not include an equality sign like equations do. For example, \(3x + 2y^2\) is an algebraic expression.
Working with algebraic expressions involves understanding how to manipulate them, including operations such as addition, subtraction, multiplication, and division.
  • Variables represent unknown values in an expression and are typically denoted by letters (e.g., \(x\), \(y\)).
  • Algebraic expressions can be complex, involving roots, fractions, and multiple variables.
To rationalize denominators in expressions with cube roots, knowing how these expressions can be rewritten and simplified is key. This manipulation often leads to easier problem-solving tactics, allowing one to proceed with additional algebraic operations without the complication of roots in the denominator.

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

Explain how to perform this multiplication: \(\sqrt{2}(\sqrt{7}+\sqrt{10})\)

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{35}{5 \sqrt{2}-3 \sqrt{5}}$$

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$-\sqrt{\frac{150 a^{3}}{b^{5}}}$$

What are conjugates? Give an example with your explanation.

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\sqrt{\frac{5 m^{4} n^{6}}{15 m^{3} n^{4}}}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.