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Chapter 1: Problem 19

Evaluate \(25-(6-2)\) and evaluate \((25-6)-2 .\) Do you think subtraction is associative?

Short Answer

Expert verified
Subtraction is not associative. Evaluated expressions give different results: 21 and 17.

Step by step solution

01

Evaluate the first expression

First, simplify the expression inside the parentheses: \(25 - (6-2)\). Calculate \(6 - 2 = 4\). So, the expression becomes \(25 - 4\). Then, calculate \(25 - 4 = 21\).
02

Evaluate the second expression

Next, simplify the second expression: \((25 - 6) - 2\). First, calculate \(25 - 6 = 19\). Then, the expression becomes \(19 - 2\). Finally, calculate \(19 - 2 = 17\).
03

Determine if subtraction is associative

Compare the results from the two expressions. The first expression evaluates to 21, while the second expression evaluates to 17. Since the results are different, subtraction is not associative. Therefore, \( \text{associative property: } (a - b) - c eq a - (b - c) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Associative Property
The associative property is a fundamental concept in mathematics, describing how the grouping of numbers affects their operation. However, it's important to note that this property applies to addition and multiplication, but not to subtraction.

For instance, in the given exercise, we evaluated two different expressions: \(25 - (6-2)\) and \((25-6) - 2\). The results were different (21 and 17, respectively). This demonstrates that changing the grouping of numbers in subtraction affects the outcome, proving that subtraction is not associative.
Subtraction
Subtraction is a basic arithmetic operation used to find the difference between two numbers. Unlike addition and multiplication, subtraction is sensitive to the order of numbers and their grouping.

In this exercise, we clearly saw how: \(25 - (6 - 2)\) gave us 21, while \((25 - 6) - 2\) resulted in 17. This highlights that the position of the parentheses changes the result, emphasizing the importance of properly grouping numbers in subtraction to get the correct answer.
Order of Operations
Understanding the order of operations is crucial when simplifying or evaluating mathematical expressions. The common acronym PEMDAS can help: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our exercise, we first resolved the expression inside the parentheses before performing the subtraction: \(6 - 2 = 4\) for \(25 - (6-2)\), and \(25 - 6 = 19\) for \((25-6) - 2\). By following the order of operations, we ensured that we evaluated the expressions correctly.
Parentheses in Equations
Parentheses play a vital role in mathematical equations as they dictate which operations should be performed first. This is especially important in operations like subtraction where the grouping of numbers can drastically change the result.

Take the original problem: \(25 - (6 - 2)\) versus \((25 - 6) - 2\). The parentheses tell us which numbers to subtract first, leading to different outcomes. This exercise clearly illustrates why understanding and correctly using parentheses is essential to accurately solving equations.

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