/*! 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 124 Express with rational denominato... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 124

Express with rational denominator \(4 /\left({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3}+1\right)\).

Short Answer

Expert verified
The simplified expression with a rational denominator is \(\frac{({ }^{3} \sqrt{9})+({ }^{3}\sqrt{3})+1}{3({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})}\).

Step by step solution

01

Identify the conjugate

Notice that we have two cube roots with subtraction in the middle in the denominator. The quadratic conjugate of the denominator, \({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3}+1\), will have the opposite sign between the two cube roots: \({ }^{3} \sqrt{9}+{ }^{3} \sqrt{3}+1\).
02

Multiply by the conjugate

Now, we'll multiply both the numerator and denominator by the conjugate. This is to eliminate the cube roots from the denominator: \[ \frac{4}{\left({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3}+1\right)} \cdot \frac{\left({ }^{3} \sqrt{9}+{ }^{3} \sqrt{3}+1\right)}{\left({ }^{3} \sqrt{9}+{ }^{3} \sqrt{3}+1\right)} \]
03

Calculate the numerator

First, we calculate the new numerator, which is just the product of 4 and each term in the conjugate: \[ 4({ }^{3} \sqrt{9})+4({ }^{3}\sqrt{3})+4 \]
04

Calculate the denominator

Next, we calculate the new denominator. To do this, use the difference of cubes formula \((a^3-b^3)=(a-b)(a^2+ab+b^2)\), considering \(\,a = {}^3\sqrt{9}\) and \(b = {}^3\sqrt{3}\). Substitute and simplify the expression: \[ (a-b)(a^2+ab+b^2) = ({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})(9 + ({ }^{3} \sqrt{9})({ }^{3} \sqrt{3}) + 3) = ({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})(12 +({ }^{3} \sqrt{27})) = 12({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3}) \]
05

Simplify the fraction

Divide numerator and denominator by 4: \[ \frac{4({ }^{3} \sqrt{9})+4({ }^{3}\sqrt{3})+4}{12({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})} = \frac{({ }^{3} \sqrt{9})+({ }^{3}\sqrt{3})+1}{3({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})} \] Now, we have the given expression with a rational denominator: \[ \frac{({ }^{3} \sqrt{9})+({ }^{3}\sqrt{3})+1}{3({ }^{3} \sqrt{9}-{ }^{3} \sqrt{3})} \]

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!

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

Show that (a) \(\quad(-8)^{2 / 3}=\left(-8^{1 / 3}\right)^{2}\) (b) \(\quad(1 / 64)^{4 / 3}=\left[(1 / 64)^{1 / 3}\right]^{4}\)

Evaluate \(16^{-3 / 4}\)

Approximate \(\sqrt{23} \times \sqrt{40}\) and \(\sqrt{23} / \sqrt{40}\)

Find the product \(\left.{ }^{3} \sqrt{(} 2 \mathrm{a}^{2} \mathrm{x}\right) \cdot \sqrt{(2 \mathrm{x})}\)

Simplify \(5 \sqrt{12}+3 \sqrt{75}\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Decision Maths

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.