Problem 47 - Prealgebra and Introductory Algebra

Question

Identify the property that justifies the statement. $$-3(9 \cdot 4)=(-3 \cdot 9) 4$$

Step-by-Step Solution

Answer

The property that justifies the given statement is the Associative Property of Multiplication.

Step 1: Identify the original equation

The original equation given is \( -3(9 \cdot 4)=(-3 \cdot 9) 4\).

Step 2: Redistribute the brackets

Rearrange the brackets to demonstrate the property: \( -3 \cdot (9 \cdot 4) = (-3 \cdot 9) \cdot 4 \).

Step 3: Identify the property

Seeing that changing the groupings of numbers in the multiplication operation doesn't change the product, we identify the property as the Associative Property of Multiplication.

Key Concepts

Multiplication Properties

When working with multiplication in mathematics, understanding its properties can help make calculations easier. The multiplication properties tell us how to operate on numbers and variables effectively. Here are some key properties:

• Commutative Property: This property states that the order in which two numbers are multiplied does not affect the product, i.e., \(a \cdot b = b \cdot a\).
• Associative Property: This suggests that the way numbers are grouped in a multiplication problem does not change the result. For instance, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Identity Property: When any number is multiplied by 1, the product is the number itself: \(a \cdot 1 = a\).
• Zero Property: Multiplying any number by zero results in a product of zero: \(a \cdot 0 = 0\).

Grasping these properties allows students to simplify complex equations and solve problems more efficiently.

Prealgebra

Prealgebra serves as the foundation of algebra, focusing on basic mathematical concepts that prepare students for higher-level math. It's an essential phase where students develop skills in manipulating numbers and recognizing patterns.
Throughout prealgebra, students learn fundamental operations such as addition, subtraction, multiplication, and division. They also explore:

• Variables: Symbols used to represent unknown numbers or values.
• Equations: Mathematical statements indicating that two expressions are equal.
• Order of Operations: The standardized sequence to solve mathematical expressions, using PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Delving into prealgebra equips students with higher analytical and problem-solving skills, forming a solid base for algebra and beyond.

Mathematical Properties

Mathematical properties are rules that apply to arithmetic operations. These properties are paramount in solving equations more efficiently and understanding the behavior of numbers.Here are a few key mathematical properties:

• Commutative Property: This applies to both addition and multiplication. Changing the order of acts does not alter the sum or the product.
• Associative Property: This is applicable in addition and multiplication, where altering the grouping does not affect the sum or product.
• Distributive Property: Allows for distributing multiplication over addition or subtraction, i.e., \(a(b + c) = ab + ac\).
• Identity and Inverse Properties: The identity property involves adding 0 or multiplying by 1 without changing a number’s value, while the inverse relates to using opposites to reach 0 or 1.

Understanding these properties helps to simplify math, making it approachable and applicable across different mathematical problems.

Distributive Property

The distributive property is a fundamental concept often used to simplify algebraic expressions and solve equations. It allows one to "distribute" multiplication across terms inside a parentheses.
This property can be expressed as \(a(b + c) = ab + ac\). Here’s a breakdown:

• Multiply the term outside the parenthesis by each term inside.
• Sum the resulting products.

For example, if we have \(3(x + 4)\), applying the distributive property gives us \(3x + 12\).
The distributive property is widely used in prealgebra to ensure expressions are manageable and helps in solving equations involving variables efficiently. Understanding this property nurtures algebraic thinking, which is crucial as students progress in mathematics.