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Chapter 1: Problem 37

Use the associative law of multiplication to write an equivalent expression. $$ 3[2(a+b)] $$

Short Answer

Expert verified
The equivalent expression is \[ 6(a+b) \].

Step by step solution

01

Understand the Associative Law of Multiplication

The associative law of multiplication states that for any numbers or variables, the grouping of these numbers does not change the product. In mathematical terms, \(a(bc) = (ab)c\). This property will be used to rewrite the given expression.
02

Identify the Groupings in the Expression

The expression is given as \(3[2(a+b)]\). Notice the current grouping is \(2(a+b)\) first and then multiplying by 3.
03

Rearrange Using the Associative Law

Applying the associative law, we can change the grouping without changing the order of multiplication. This will give us \[ 3 \times 2 \times (a+b) \]. The new grouping is \( (3 \times 2)(a+b) \).
04

Simplify the Multiplication

Now, multiply the constants 3 and 2 together to simplify the expression. This gives \[ 6(a+b) \].

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Key Concepts

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

Properties of Multiplication
Multiplication in mathematics has several important properties.
Understanding these properties is essential as they make solving problems easier and more intuitive. For our exercise, we focused on the associative property, but let's also touch on a few others:
  • Associative Property: This property states that the grouping of numbers does not affect their product. Mathematically, this is expressed as \(a(bc) = (ab)c\).
  • Commutative Property: This one tells us that the order in which numbers are multiplied does not change the product: \(ab = ba\).
  • Distributive Property: It combines addition and multiplication. It shows that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the results: \(a(b + c) = ab + ac\).

In the exercise given, we applied the associative property, which allowed us to regroup the numbers and simplify the expression.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and arithmetic operations. In our exercise, the expression provided was \(3[2(a+b)]\). This involves numbers (3 and 2) and a variable-based expression (a+b).
The variables in algebraic expressions represent unknown values and can take different values. Here, \(a\) and \(b\) could be any number. Understanding how to manipulate these expressions using different properties and laws, like the associative law, is crucial in algebra.
It's important to remember:
  • An expression can be simplified by combining like terms or using properties of operations.
  • The parentheses indicate which operations should be performed first.
This fundamental understanding of algebraic expressions helps break down complex problems into more manageable parts.
Simplification
Simplification is the process of reducing an expression to its simplest form. In our exercise, we simplified \(3[2(a+b)]\) by using the associative property.

The steps for simplifying effectively include:
  • Identifying Groupings: Recognize how the terms are grouped initially.
  • Applying Properties: Use properties like the associative law to rearrange the terms.
  • Combining Like Terms: Multiply constants and simplify the expression.
Step by step, we transformed \(3[2(a+b)]\) into \(6(a+b)\). This final form is more straightforward and easier to work with for further operations or evaluations.

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Most popular questions from this chapter

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