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Chapter 1: Problem 725

Use the associative property to simplify. $$ (y+12)+28 $$

Short Answer

Expert verified
The simplified expression is \(y + 40\).

Step by step solution

01

Identify the Expression

The given expression is \( (y + 12) + 28 \). We need to use the associative property to simplify this.
02

Apply the Associative Property

The associative property of addition states that the way in which numbers are grouped does not change their sum. So, we can rewrite the expression by regrouping: \((y + 12) + 28 = y + (12 + 28)\).
03

Simplify the Grouped Terms

Now, add the numbers inside the parentheses: \(12 + 28 = 40\). So, the expression becomes \(y + 40\).
04

Rewrite the Simplified Expression

The simplified expression is \(y + 40\).

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Key Concepts

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

associative property of addition
The associative property of addition is a key concept in mathematics. It states that when adding numbers, the grouping of the numbers does not affect their sum.
For example, if you have the expression \(a + (b + c)\), it can be rewritten as \((a + b) + c\) and you will get the same result.
This property is very useful, especially when dealing with algebraic expressions, as it allows us to regroup terms to simplify calculations.
simplification
Simplification means reducing an equation or expression to its simplest form. Using properties like the associative property makes this easier by reorganizing the terms.
For instance, in the given expression from the exercise \( (y + 12) + 28 \), you can use the associative property to group the numbers in a different way.
This helps to combine like terms and make the expression more compact and easier to work with.
In the example, we first regroup the terms to \( y + (12 + 28) \).
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. They do not have an equality sign like equations.
Understanding how to manipulate these expressions using properties like the associative property can greatly simplify your work.
In the given exercise, we start with the algebraic expression \( (y + 12) + 28 \).
Through simplification, using the associative property, we eventually arrive at a much simpler expression \ y + 40 \.
This step-by-step manipulation is crucial for solving more complex problems in algebra.

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