Problem 31 - Prealgebra

Question

Find each sum or product. Explain your reasoning. $$125 \cdot 4 \cdot 0$$

Step-by-Step Solution

Answer

The product is 0.

Step 1: Identify the Elements in the Expression

The expression given is a product of three numbers: 125, 4, and 0. Our goal is to find the product of these three numbers.

Step 2: Apply the Zero Property of Multiplication

According to the zero property of multiplication, any number multiplied by zero results in zero. Identify the presence of zero in the product: \(125 \cdot 4 \cdot 0\).

Step 3: Simplify the Expression

Since one of the factors in the product is zero, by the zero property of multiplication, the entire expression evaluates to zero without further calculations needed.

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation aimed at finding the total of one number taken a specific number of times. The simplest way to understand it is as repeated addition. For example, when you multiply 4 by 3, you're essentially adding 4 three times:

• 4 + 4 + 4 = 12

Multiplication uses the symbol "\( \cdot \)," or sometimes simply "x" in informal settings, to connect numbers called factors, each representing quantities in the operation. In our problem, the numbers 125, 4, and 0 are multiplied together:

• \(125 \cdot 4 \cdot 0\)

The purpose of multiplication is to simplify calculations and expressions into their streamlined numeric total. This makes it easier to handle larger numbers and more complex operations in everyday math.

Mathematical Expression

A mathematical expression is a combination of numbers, variables, and operators creating a "sentence" that requires evaluation or simplification. For instance, the expression in our context is:

• \(125 \cdot 4 \cdot 0\)

Such expressions can involve adding, subtracting, multiplying, or dividing numbers or other expressions. They are essential because they allow us to communicate complex mathematical ideas in a concise form. Within a mathematical expression, operators like addition or multiplication dictate how to process the numbers involved. Evaluating an expression calls for understanding and applying mathematical rules to find the result or value. In our problem, multiplication is the primary operation connecting all numbers in the expression.

Simplification

Simplification in mathematics means reducing an expression to its simplest form. This makes it easier to understand and work with. With our example, the multiplication sequence includes a zero, making it prime for simplification under the zero property of multiplication. Typically, identifying key components such as zero is an important step, because it can drastically affect the outcome. Through simplification, we apply mathematical rules to minimize the expression while maintaining its equivalency:

• Since one factor is zero, we directly simplify the expression to zero: \(125 \cdot 4 \cdot 0 = 0\).

This reduction is crucial when solving problems efficiently, especially in complex calculations where certain properties can be leveraged for quick answers.

Number Properties

Number properties are rules governing the behavior of numbers in operations, ensuring consistent results in mathematics. One key property applied here is the Zero Property of Multiplication. It states that any number multiplied by zero results in zero. Taking our example, the presence of zero in the expression \(125 \cdot 4 \cdot 0\) makes applying this property straightforward:

• As zero is a factor, the entire product simplifies immediately to zero: \(0\).

Using such number properties simplifies computations and confirms correctness in mathematical problem-solving. Recognizing and applying these rules in various situations enables easier handling of larger, more complex expressions. Besides the zero property, other properties like the associative and commutative properties of multiplication provide additional framework for managing arithmetic operations.