Problem 13 - Precalculus Mathematics for Calculus

Question

State the property of real numbers being used. $$2 x(3+y)=(3+y) 2 x$$

Step-by-Step Solution

Answer

Commutative Property of Multiplication.

Step 1: Identify the Expressions

Look at the expression on both sides of the equation. We have \(2x(3+y)\) on the left side and \((3+y)2x\) on the right side.

Step 2: Recognize the Property

The equation \(2x(3+y) = (3+y)2x\) involves a rearrangement of the factors \(2x\) and \(3+y\). This is identifying a specific property of real numbers.

Step 3: State the Property

This rearrangement of factors showcases the **Commutative Property of Multiplication**, which states that changing the order of the factors does not change the product.

Key Concepts

Properties of Real Numbers

Real numbers play a fundamental role in mathematics and have specific properties that help us understand and work with them more effectively. One of the key properties is **associative**, which suggests the grouping of numbers does not affect their sum or product.
Another is **distributive**, relating to the distribution of multiplication over addition or subtraction. However, the property in focus today is the **commutative property**. This property relates specifically to addition and multiplication, and it is at the heart of the exercise we are examining.
The commutative property states:

• For addition: \( a + b = b + a \)
• For multiplication: \( a \cdot b = b \cdot a \)

This property tells us that it doesn't matter in which order you add or multiply numbers—the result will be the same. This makes calculations flexible and allows us to rearrange terms during algebraic manipulations.

Multiplication in Algebra

Algebra often involves the multiplication of numbers and variables, and understanding how to use the commutative property effectively can simplify this process. When you multiply expressions in algebra, such as \(2x(3+y)\), it's essential to recognize that rearranging their order yields the same result thanks to the commutative property.
In algebra, expressions can get complex, with many different factors and parentheses. This is where knowing properties like commutative comes into play.
Consider the expression in the exercise: \(2x(3+y)\). Here, it demonstrates a multiplication of a constant \(2x\) and a binomial \((3+y)\). By rearranging it to \((3+y)2x\), no value is lost or changed. This demonstrates how algebraic multiplication remains consistent under factor rearrangement due to the commutative property.

Rearrangement of Factors

Rearranging factors focuses on the concept that, during multiplication, the order of factors does not affect the outcome or value of the expression. This is a direct application of the commutative property, which applies to all real numbers.
In practice, this means that if you have an expression such as \( a \cdot b \), you can rearrange it to \( b \cdot a \) without changing its value.
In algebra, this rearrangement can make solving or simplifying expressions more convenient, especially when dealing with complex expressions. Recognizing that \((3+y)2x = 2x(3+y)\) can help streamline calculations and solve problems more quickly. By constantly practicing these rearrangements, one can develop a more intuitive understanding of algebraic structures and enhance problem-solving skills.