Problem 22 - Elementary and Intermediate Algebra

Question

Use the commutative law of multiplication to write an equivalent expression. $$ 4 x $$

Step-by-Step Solution

Answer

The equivalent expression is \( x \times 4 \).

Step 1: Understand the Commutative Law of Multiplication

The commutative law of multiplication states that changing the order of factors does not change the product. Mathematically, it is expressed as: \[ a \times b = b \times a \]

Step 2: Identify the Factors

In the given expression \( 4x \), the factors are 4 and \( x \).

Step 3: Apply the Commutative Law

By applying the commutative law, swap the order of the factors. Therefore, \( 4x \) becomes \( x \times 4 \).

Step 4: Write the Equivalent Expression

The equivalent expression, after applying the commutative law, is \( x \times 4 \).

Key Concepts

commutative property

The commutative property is a fundamental principle in mathematics. It tells us that the order in which we add or multiply numbers does not matter. This property holds for both addition and multiplication.
For multiplication, it means that if you have two numbers (or variables), swapping them won't affect the result. For example, the commutative property states \( a \times b = b \times a \).
In our exercise, we use this property to show that \( 4 \times x \) is the same as \( x \times 4 \). This concept is useful in simplifying expressions and solving equations. It's a building block for more complex algebraic manipulations.

multiplication

Multiplication is one of the basic operations in arithmetic and algebra. It combines groups of equal sizes. When we multiply, we make repeated addition easy and concise.
For instance, \( 4 \times x \) means that we have 4 groups of x. Multiplication has special properties:

• Commutative: \( a \times b = b \times a \)
• Associative: \( (a \times b) \times c = a \times (b \times c) \)
• Distributive: \( a \times (b + c) = (a \times b) + (a \times c) \)

These properties make algebraic manipulation possible. In our exercise, the commutative property simplifies \( 4 \times x \) to \( x \times 4 \).

algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It helps in expressing general mathematical relationships and forming equations.
In algebra, we often represent numbers with letters (like x) and perform operations on them. The commutative property is one of the key principles in algebra. It ensures that the order in which we multiply does not affect the outcome.
In the given exercise, algebra helps us understand that the expression \( 4x \) can be rewritten as \( x \times 4 \). This flexibility in manipulating expressions is what makes algebra so powerful and essential in solving complex problems.