Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of multiplication to write an equivalent algebraic expression. $$9 x$$

Step-by-Step Solution

Answer

The equivalent expression of \(9x\) using the commutative property of multiplication is \(x \cdot 9\).

Step 1: Understand the Exercise

The exercise asks to rewrite the algebraic expression \(9x\) using the commutative property of multiplication. Commutative property states that changing the order of factors does not change the product.

Step 2: Apply the Commutative Property

Applying the commutative property to \(9x\), interchanging the order of multiplication, gives \(x \cdot 9\).

Step 3: Write Down Your Final Answer

The equivalent expression using the commutative property is \(x \cdot 9\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a way to express mathematical ideas using variables, numbers, and operations. When you see something like \(9x\), the expression consists of a numerical coefficient (9) and a variable (\(x\)). The variable can stand for any number, which is why these expressions are so versatile.
Understanding how to manipulate algebraic expressions is crucial. It lets you solve equations and model real-world situations.
For example, if \(x\) represented apples, then \(9x\) would mean nine apples. Here, comprehension of algebraic expressions leads to practical problem-solving.

Properties of Multiplication

The properties of multiplication are rules that make it easier to work with numbers. One essential rule is the commutative property, which states that the order of factors does not affect the product. For example, \(a \times b = b \times a\).
By applying this property, you can switch around the numbers and variables without changing their multiplication result.

• This makes calculations more flexible and streamlined.
• It helps simplify complex expressions or equations.

In our example, the commutative property is used to show that \(9x\) is equivalent to \(x \times 9\). Remember, properties of multiplication form the groundwork for effective problem-solving in math.

Mathematical Concepts

Mathematical concepts are foundational ideas that enrich our understanding of math as a whole. Comprehension of these concepts helps in problem-solving and logical thinking.
The commutative property of multiplication is a basic mathematical concept and demonstrates the flexibility of operations such as multiplication.
When you recognize that numbers can be rearranged but still produce the same product, you're using abstract thinking that is valuable not just in math but in everyday life.

• It encourages systematic approaches to problem-solving.
• Enables solving real-world problems with mathematical models.

By mastering these concepts, you build a toolkit that makes tackling math challenges much more manageable.