Question

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

Short answer

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

Identify the given expression

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

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).

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.

Key concepts

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.