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Chapter 2: Problem 1

Apply the associative property to expression, and then simplify the result. 5(4 a)

Short Answer

Expert verified
The expression simplifies to \(20a\).

Step by step solution

01

Identify the Associative Property

The associative property refers to the way numbers are grouped in multiplication or addition; it states that the grouping does not change the result. For multiplication, this property implies: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \). In this exercise, the expression is \(5(4a)\), which can be reorganized using the associative property.
02

Apply the Associative Property

Using the associative property for the multiplication, rewrite the expression \(5(4a)\) as \((5 \cdot 4) \cdot a\). This changes the grouping of the numbers in the multiplication without changing the result of the expression.
03

Simplify the Expression

Now perform the multiplication inside the parentheses first. Calculate \(5 \cdot 4 = 20\). Substitute this back into the expression: \(20 \cdot a\). Therefore, the simplified expression is \(20a\).

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Key Concepts

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

Multiplication Properties
When we talk about multiplication properties, we primarily focus on how numbers behave during multiplication. One of the fundamental properties is the associative property of multiplication. This property states that when you multiply three or more numbers, the way you group them does not change the result. In mathematical terms, this means
  • \((a \cdot b) \cdot c = a \cdot (b \cdot c) \).
  • It allows flexibility in grouping numbers.
This concept is quite forgiving, letting you choose the most convenient order to perform multiplications. It makes calculations more versatile, especially when dealing with larger numbers. For instance, by grouping numbers we are comfortable with, calculations become easier, even outside of academic exercises.
Expression Simplification
Expression simplification is the process of reducing an expression to its simplest form. By applying specific rules and properties from math, we aim to make the expression as straightforward as possible. In the context of multiplication, we often use properties like the associative property to rearrange and simplify the expression.
  • You can think of it as cleaning up a math problem to express it more neatly.
  • For example, transforming \(5(4a)\) into \(20a\) cleans up clutter.
When simplifying, start from grouping numbers that are most manageable, perform any straightforward multiplications, and finally write down the neat result. This practice not only makes the expression simpler, but it also proves essential in solving more complex algebra problems.
Prealgebra Principles
In prealgebra, principles like the associative property form the backbone of understanding mathematical operations. This stage is crucial as it introduces students to how numbers interact with one another through various operations. Prealgebra focuses on:
  • Establishing a strong foundation for handling variables and expressions.
  • Teaching properties that simplify problem-solving processes.
  • Encouraging a logical progression in steps to solve equations or simplify expressions.
These fundamental concepts serve as stepping stones for more advanced algebraic operations. By mastering prealgebra principles, students become well-equipped to tackle more challenging math courses as they have honed their ability to manipulate and simplify different types of expressions and equations.

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Most popular questions from this chapter

Simplify each of the following. $$(7+1)(7-1)$$

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-29-53-37$$

Translate each of the following and simplify the result. Find the difference of \(-5\) and \(-1\)

Rewrite the following using the commutative property of multiplication. $$7(x)=$$

Perform the indicated operations. $$(6+1)(9-4)$$
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