/*! 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 32 simplify by removing all possibl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 32

simplify by removing all possible factors from the radical. $$ \sqrt[3]{\frac{16}{27}} $$

Short Answer

Expert verified
The simplified value of \( \sqrt[3]{\frac{16}{27}} \) is \( \frac{2\sqrt[3]{2}}{3} \).

Step by step solution

01

Identify Perfect Cubes

Locate the perfect cubes in both the numerator and denominator. The perfect cube in the denominator is \( 27 = 3^3 \) and the perfect cube in the numerator is \( 16 = 2^4 \) , which contains the perfect cube \( 8=2^3 \) as a factor.
02

Simplify the Expression

Next, break down the original expression into the discovered perfect cubes. Break down \( 16 = 8 \cdot 2 \), which simplifies to \( \sqrt[3]{8 \cdot 2} \). And then, we write 27 as \( 3^3 \) so our expression becomes \( \sqrt[3]{(2^3 \cdot 2) / 3^3} \).
03

Apply Cube Root

Now, apply the cube root on the simplified numerator and denominator. That turns into \( 2 \cdot \sqrt[3]{2} / 3 \).

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.

Cube Roots
The cube root operation is one of the more fascinating concepts in mathematics that needs to be clearly understood. A cube root, denoted as \( \sqrt[3]{x} \), essentially means finding a number that, when multiplied by itself three times, results in \( x \). Understanding Cube Roots:
  • Unlike square roots, which look for a number squared, cube roots search for a number cubed.

  • For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).

  • We express this operation as \( \sqrt[3]{8} = 2 \).
Cube roots are used to simplify expressions involving radicals by reducing them to a simpler form. In our example, finding the cube root of the given expression \( \sqrt[3]{\frac{16}{27}} \) means identifying factors of the numerator and denominator that are perfect cubes, which simplifies the radical considerably.
Perfect Cubes
Perfect cubes are numbers that result from cubing a whole number. Recognizing these is crucial when simplifying cube roots. What are Perfect Cubes?
  • The first few perfect cubes include numbers such as 1, 8, 27, 64, and so on, because \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), etc.

  • To simplify a radical expression like \( \sqrt[3]{16/27} \), you need to identify any perfect cubes within that fraction.
In our exercise, 27 is already a perfect cube (\( 3^3 \)). In the numerator, 16 includes a smaller perfect cube, which is 8 (\( 2^3 \)). These identifications are what allow us to simplify the equation further, by taking out the factor \( 8 \) and leaving the radical expression \( \sqrt[3]{2} \).
Denominator Simplification
Simplifying the denominator is an important step that often makes a radical expression easier to work with. It ensures that we have whole number denominators which are clean and simple. Steps to Simplify the Denominator:
  • Since 27 is a perfect cube (\( 3^3 \)), finding its cube root directly gives us the whole number 3.

  • This simplifies the fraction from having a complex denominator to a simple whole number.
After identifying and extracting the perfect cube from the denominator \( 27 \), the original expression takes a much clearer form. This gives us \( 2 \times \sqrt[3]{2} / 3 \), displaying how powerful denominator simplification can be in handling radical expressions efficiently.

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 domain of the given expression. $$ \sqrt{x-4} $$

Rationalize the numerator or denominator and simplify. $$ \frac{2 x}{5-\sqrt{3}} $$

Rationalize the numerator or denominator and simplify. $$ \frac{3}{\sqrt{21}} $$

a certificate of deposit has a principal of P and an annual percentage rate of r (expressed as a decimal) compounded n times per year. Enter the compound interest formula $$ A=P\left(1+\frac{r}{n}\right)^{N} $$ into a graphing utility and use it to find the balance after \(N\) compoundings. $$ P=\$ 7000, \quad r=5 \%, \quad n=365, \quad N=1000 $$

Perform the indicated operations and rationalize as needed. $$ \frac{\frac{\sqrt{4-x^{2}}}{x^{4}}-\frac{2}{x^{2} \sqrt{4-x^{2}}}}{4-x^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.