Problem 50 - Prealgebra and Introductory Algebra

Question

Use the given property of multiplication to complete the statement. The Multiplication Property of Zero \(-81 \cdot ?=0\)

Step-by-Step Solution

Answer

The missing number is 0.

Step 1: Understand the Multiplication Property of Zero

The Multiplication Property of Zero states that any number multiplied by zero will give a product of zero. It can be expressed as \(a \cdot 0 = 0\) or \(0 \cdot a = 0\), where 'a' is any real number.

Step 2: Apply the Multiplication Property of Zero to the Provided Statement

The statement given by the exercise is \(-81 \cdot ? = 0\). Using the Multiplication Property of Zero, replace the question mark with zero.

Key Concepts

Basic Multiplication

Multiplication is one of the fundamental operations of arithmetic, much like addition, subtraction, and division. It involves combining quantities in groups. For example, if you have 3 groups of 4 apples, how many apples do you have in total? You get the answer by multiplying 3 and 4, which gives you 12. In this case, the number 3 tells you how many groups, and 4 tells you how many are in each group, so 3 times 4 is 12. The operation is expressed as:

• 3 multiplied by 4 equals 12 (3 \( \times \) 4 = 12)
• Also written as 3 \( \cdot \) 4 = 12, following the format used in your algebra classes.

Multiplication is commutative and associative, meaning the order or grouping of numbers does not change the product. Understanding this basic principle helps when learning more complex operations like properties of numbers.

Zero Property

The Multiplication Property of Zero is an important principle in mathematics, especially in algebra. It simplifies computations and helps verify solutions. When you multiply any number by zero, the result is always zero, symbolized by the formula:

• \( a \cdot 0 = 0 \)
• This holds true for any real number 'a.'

This property keeps equations neat and confirms that adding more zeroes will still keep the product zero. For example, in the exercise \( -81 \cdot ? = 0 \), we directly know \(?\) is 0. Why? Because multiplying -81, or any number, by zero results in zero.This property is particularly useful when solving equations because it quickly sets terms containing zero to zero without any computation required.

Algebraic Properties

Algebraic properties are foundational rules used to manipulate mathematical expressions and equations. The Multiplication Property of Zero is one such rule. Others include:

• Commutative Property: This states that \( a \cdot b = b \cdot a \), meaning the order of numbers doesn't affect the result.
• Associative Property: It claims \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \), which means grouping doesn't change the product.
• Distributive Property: It shows how multiplication interacts with addition, such as \( a \cdot (b + c) = a \cdot b + a \cdot c \).

In algebra, knowing these properties allows you to simplify and solve equations more efficiently. The zero property is particularly highlighted in this context, ensuring expressions reduce appropriately when zero is a factor. Therefore, these properties collectively support effective problem-solving and deeper understanding in math.