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Chapter 5: Problem 4

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

Short Answer

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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.

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