Problem 86 - Algebra 2

Question

Name the property of real numbers illustrated by each equation. $$ 4(3 x)=(4 \cdot 3) x $$

Step-by-Step Solution

Answer

The property illustrated is the associative property of multiplication.

Step 1: Examine the Equation

We need to identify which property of real numbers is illustrated by: \(4(3 x)=(4 \cdot 3) x\).

Step 2: Recall Properties of Real Numbers

The key properties are:

• Commutative Property: \(a + b = b + a\) or \(a \cdot b = b \cdot a\)
• Associative Property: \((a + b) + c = a + (b + c)\) or \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
• Distributive Property: \(a(b + c) = ab + ac\)
• Identity Property: \(a + 0 = a\) or \(a \cdot 1 = a\)
• Inverse Property: \(a + (-a) = 0\) or \(a \cdot \frac{1}{a} = 1\)

Step 3: Identify the Property

The property illustrated is the associative property of multiplication.

Key Concepts

Properties of Real Numbers

When dealing with the properties of real numbers, it's essential to understand how these properties allow us to manipulate and simplify mathematical expressions. One such property, as illustrated by the equation \(4(3x) = (4 \cdot 3)x\), is the associative property of multiplication. This crucial property tells us that when three or more numbers are multiplied together, the grouping of these numbers doesn't affect the product. In other words, rearranging the parentheses in a multiplication equation doesn't change the result. For instance, \(a(bc) = (ab)c\) for any real numbers \(a\), \(b\), and \(c\).

Properties like commutativity (\(ab = ba\)), identity (\(a\cdot1 = a\)), and distributivity (\(a(b + c) = ab + ac\)) are additional properties of real numbers that enable us to perform algebraic operations with confidence. These properties form the foundation of our number system and are ingrained in the rules of arithmetic we use every day.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and the operations of addition, subtraction, multiplication, division, and exponentiation. A key point to remember is that an algebraic expression represents a value that can change if the variables within it change.

Understanding how to manipulate these expressions using properties of real numbers is a cornerstone of algebra. When simplifying expressions, we often rely on the associative property of multiplication, as seen in the equation \(4(3x) = (4 \cdot 3)x\), to regroup terms for easier computation. By mastering the use of these properties, students become better equipped to handle more complex problems, such as solving equations, factoring, expanding, and simplifying algebraic expressions.

Algebra 2

Algebra 2 takes the lessons learned in earlier algebra courses to the next level, diving deeper into the properties and operations of algebraic expressions and functions. Topics often explored include quadratic equations, exponential and logarithmic functions, sequences and series, probability and statistics, and many more advanced concepts. The associative property of multiplication is one example of a foundation concept from Algebra 1 that remains essential throughout Algebra 2. This property, along with others, becomes even more critical as students tackle higher-level algebraic problems involving polynomial expressions, complex numbers, and matrix operations.

By thoroughly understanding properties like the associative property, students can simplify and solve sophisticated equations more efficiently. For example, mastering this property aids in polynomial multiplication and the application of functions, setting the stage for further study in mathematics, including calculus.