Problem 35 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The equivalent expression is \( (2a)b \)

Step 1: Understand the Associative Law of Multiplication

The associative law of multiplication states that for any numbers or variables, the grouping of the factors does not change the product. Mathematically, it can be expressed as \[ (a \times b) \times c = a \times (b \times c) \]

Step 2: Identify the factors

In the given expression, 2 and \(ab\) are the factors being multiplied in the order of \(2 \times (a \times b)\).

Step 3: Apply the Associative Law

Rearrange the factors using the associative law. The expression becomes \[ (2 \times a) \times b \]

Step 4: Simplify the Expression

Simplify the expression if needed. In this case, the simplified form is still \[ (2a)b \]

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that focuses on variables and the rules for manipulating these variables. In algebra, we often work with expressions and equations where symbols represent numbers.

In the given exercise, variables such as 'a' and 'b' are used to represent unknown values. The use of variables allows us to create general expressions and solve problems in a more flexible way.

When dealing with variables, it's important to understand the properties and laws that govern their operations, such as the associative law of multiplication. This law helps us see that the way we group variables or numbers when multiplying does not affect the final result.

Multiplication properties

Multiplication has several fundamental properties that make calculations easier and more understandable. Some of these include the commutative property, the associative property, and the distributive property.

Commutative Property: This property states that the order in which you multiply numbers or variables does not change the product. Mathematically, this is written as \( a \times b = b \times a \).

Associative Property: This property highlights that the grouping of numbers or variables does not change the product. It can be expressed as \( (a \times b) \times c = a \times (b \times c) \). For example, in the given exercise, \(2(a b)\) can be rearranged to \( (2a) b \) without changing the result.

Distributive Property: This property states that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding the products. Mathematically, it is expressed as \( a \times (b + c) = ab + ac \).

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as addition, subtraction, and multiplication) that represent a particular value.

Let's break down the given expression \( 2(a b) \):

• '2' is a constant, meaning it is a fixed value.
• 'a' and 'b' are variables, meaning they can represent different numbers.
• '(a b)' means that 'a' and 'b' are being multiplied together first.

According to the associative law of multiplication, we can rewrite this expression as \( (2a) \times b \). This does not change the overall product because the law allows us to regroup the factors.

Understanding how to manipulate and simplify mathematical expressions is crucial in algebra and helps in solving more complex equations efficiently.