/*! 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 4 Explain why no cancellation is p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Explain why no cancellation is possible in the expression \(\frac{a+2 b}{a-2 b}\).

Short Answer

Expert verified
Question: Explain why no cancellation is possible in the given expression \(\frac{a+2 b}{a-2 b}\). Answer: No cancellation is possible in the given expression because there is no common factor between the numerator and the denominator. The numerator contains the terms \((a + 2b)\) and the denominator contains the terms \((a - 2b)\). Both expressions cannot be factored any further, and there is no shared factor between them that would allow for cancellation or simplification.

Step by step solution

01

Identify the given expression

The given expression is \(\frac{a+2 b}{a-2 b}\).
02

Examine the numerator and the denominator for common factors

The numerator contains the terms \((a + 2b)\) and the denominator contains the terms \((a - 2b)\). Looking at these terms, we can see that although both have \(a\) and \(2b\), they have a different operation between them (addition in the numerator, and subtraction in the denominator).
03

Explain why no cancellation or simplification is possible

In order to cancel or simplify a fraction, the numerator and denominator must have a common factor that divides both without leaving any remainder. In the given expression, there is no common factor between the numerator and the denominator that would allow for cancellation. The lack of a common factor can be shown as follows: In the numerator, we have \((a + 2b)\). This expression cannot be factored any further. In the denominator, we have \((a - 2b)\). This expression also cannot be factored any further. Since no factor is present in both the numerator and the denominator, there is no possible cancellation between the terms.

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.

Numerator and Denominator
Every fraction is composed of two main parts: the numerator and the denominator. The numerator is the top part of the fraction, while the denominator is the bottom part.
For example, in the fraction \(\frac{a+2b}{a-2b}\), the numerator is \(a + 2b\) and the denominator is \(a - 2b\). These components serve different roles: the numerator shows how many parts we have, and the denominator indicates the total number of equal parts that make up a whole.
It is important to understand that both the numerator and denominator are integral in determining the value of a fraction. Changing either alters the fraction's overall value significantly.
Common Factors
Common factors are numbers or expressions that are shared by two or more numbers or expressions. In terms of fractions, simplifying involves finding and dividing out common factors in the numerator and denominator.
A key part of simplifying any fraction is examining whether the numerator and the denominator have any common factors. If they have one, you can simplify the fraction by dividing both parts by this factor.
  • In this exercise, however, the expressions \(a+2b\) and \(a-2b\) don't have common factors.
  • The terms look similar but the operations - addition in the numerator and subtraction in the denominator - make them different.
This is why no cancellation is possible.
Operations in Algebra
Operations in algebra involve addition, subtraction, multiplication, and division. The order and type of operations used are crucial, especially when dealing with expressions.
In the expression \(a+2b\), addition is used, combining \(a\) and \(2b\). In contrast, \(a-2b\) uses subtraction. These operations fundamentally affect the expressions, meaning they are not equivalent and cannot cancel each other out.
  • Understanding and identifying operations ensures clarity in whether simplification is possible.
  • Often, the type of operation plays a significant role in fraction simplification.
However, in this exercise, the different operations prevent finding a common factor for simplification.
Factoring Expressions
Factoring is the process of breaking down expressions into simpler 'factors' that, when multiplied together, give back the original expression. Determining whether an expression can be factored is key to simplifying fractions.
However, the given expressions \(a + 2b\) and \(a - 2b\) are already as simple as they can be - they cannot be broken down further into simpler factors.
  • Factoring aids in simplifying fractions by revealing common factors between numerator and denominator.
  • Yet, when expressions cannot break down any further, simplification through cancelation is not achievable.
Thus, understanding the factoring possibilities helps to conclude why no common cancellation occurs in such cases.

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

Simplify each of the following: (a) \(\frac{m^{-7}}{m^{-4}}\) (b) \(\left(3 a b^{2} c\right)^{3}\) (c) \(\left.y^{3} \times y^{-2} \times x^{7} \times x^{5} \times x^{-3}\right)\)

Remove the square brackets from $$ \frac{1}{\omega}\left[\frac{s \omega}{s^{2}+\omega^{2}}-s\right] $$

Reduce each of the following fractions to its simplest form: (a) \(\frac{4}{13}\) (b) \(\frac{15}{20}\) (c) \(\frac{15}{35}\) (d) \(\frac{11}{12}\)

Remove the brackets from the given expression: \(\left(x^{2}+2\right)(3 x)\)

Simplify \(\frac{5 x}{25 x+10 y}\)
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Modern Physics

Read Explanation

Quantum Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Dynamics

Read Explanation

Classical Mechanics

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.