Problem 17 - Algebra and Trigonometry

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: Understanding the Operation

Let's first consider what operation we are performing here. We have multiplication and addition inside the parentheses, but we should focus on the order in which the numbers and variables are being multiplied.

Step 2: Analyzing the Expression

The given expression is \(2x(3+y) = (3+y)2x\). If we observe closely, the numbers and expressions are swapped in their order between the left and right side of the equation.

Step 3: Identifying the Property

The property that states you can change the order of multiplication without affecting the outcome is the **Commutative Property of Multiplication**. This property asserts that for any two numbers or expressions \(a\) and \(b\), the equation \(ab = ba\) holds true.

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics. They include all the numbers that can be found on the number line. This group encompasses:

• Natural numbers (1, 2, 3, ...),
• Whole numbers (0, 1, 2, ...),
• Integers (..., -2, -1, 0, 1, 2, ...),
• Rational numbers (fractions like \( \frac{1}{2} \) or \(-3\frac{5}{7} \)), and
• Irrational numbers (like \(\pi\) and \(\sqrt{2}\)).

Real numbers can be positive, negative, or zero. They are used everywhere in everyday life and form the basis for more advanced mathematical concepts. When we perform operations such as addition, subtraction, multiplication, or division, we typically work within the realm of real numbers. Understanding these numbers is key to grasping algebraic concepts, including the properties like the Commutative Property of Multiplication.

Multiplication Property

The multiplication property in mathematics refers to rules that define how multiplication interacts with numbers and expressions. One of the most important aspects is the **Commutative Property of Multiplication**. It states that for any two real numbers or algebraic expressions:

• \( a \cdot b = b \cdot a \)

This rule means that changing the order of the factors does not change the product. For instance, in our example, the expressions \(2x(3+y)\) and \((3+y)2x\) have been reordered due to this property, yet the product remains the same.
Another related property is the Associative Property of Multiplication, which focuses on how numbers can be grouped. Remember, these properties are essential for simplifying and solving algebraic expressions and equations efficiently.

Algebraic Expressions

Algebraic expressions blend numbers, variables, and operation signs (like plus and minus) into mathematical terms. These expressions can range from something as simple as \(x+2\) to more complex ones like \(3a^2 + 4a - 5\).
An algebraic expression becomes easier to handle, solve, or simplify when we apply mathematical properties such as the Commutative Property of Multiplication. For instance, in the expression \(2x(3+y) = (3+y)2x\), this property allows us to rearrange terms in a way that may simplify the computation or better align the expression with other terms.
Understanding how to use these properties not only helps in solving problems but also builds a solid foundation for learning more advanced algebra and higher-level mathematical concepts.