Problem 108 - Elementary and Intermediate Algebra

Question

Use the associative property of multiplication to find each product. $$ -1.5(-100 \cdot 4) $$

Step-by-Step Solution

Answer

The product is 600.

Step 1: Identify the Expression

The expression we need to solve is \(-1.5(-100 \cdot 4) \).We will use the associative property of multiplication to simplify this, which states that the grouping of factors does not change the product.

Step 2: Rearrange Using the Associative Property

According to the associative property, we can regroup the numbers:\[(-1.5 \cdot -100) \cdot 4\].This means the new grouping is \((-1.5 \cdot -100)\) first, then multiply the result by 4.

Step 3: Calculate Each Group

First calculate the expression within the parentheses: \[ -1.5 \cdot -100 = 150 \].Note that multiplying two negative numbers results in a positive number.

Step 4: Find the Product

Now multiply the result from the previous step by 4:\[150 \cdot 4 = 600\].This is the final product.

Key Concepts

Understanding Algebraic Expressions

An algebraic expression involves a combination of numbers, operators (such as +, -, *, /), and at times, variables. These expressions are fundamental to algebra and mathematics as they allow us to represent real-world scenarios in a mathematical context. When solving algebraic expressions, it’s crucial to follow the order of operations and properties associated with numbers, such as the associative and commutative properties.
The exercise above requires understanding how to manipulate and rearrange expressions using these properties. Specifically, the associative property of multiplication indicates that how we group numbers in multiplication does not affect the product: \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\). This principle is key in simplifying seemingly complex expressions.
Algebraic expressions help sharpen problem-solving skills by fostering logical thinking and enhancing pattern recognition. It is through these expressions that students learn to translate real-world problems into solvable mathematical models.

Multiplying Negative Numbers

Multiplying negative numbers might initially seem confusing, but there's a simple rule to make it easier: multiplying two negative numbers always results in a positive product. This is because negative numbers denote a direction on the number line opposite to positive numbers. When you multiply two negative numbers, you are effectively reversing direction twice, which brings you back to the positive side.

• For instance, consider the expression: \(-1.5 \cdot -100\). Here, multiplying \(-1.5\) by \(-100\) yields \(150\).
• If one of the numbers is positive and the other negative, the product will be negative as shifting direction only once keeps you negative.

Understanding these rules is essential for tackling more complex algebraic expressions, ensuring students perform operations accurately and confidently.

Enhancing Problem-Solving in Mathematics

Problem-solving in mathematics isn't just about finding the right answer; it's about understanding the path to the solution. This includes recognizing patterns, employing mathematical principles, and knowing which properties and rules apply to a given expression. In our exercise, we utilized the associative property to simplify the multiplication, showcasing a strategic approach to solving problems.
To enhance problem-solving:

• Break down the problem into manageable parts.
• Look for properties that can simplify calculations.
• Review steps regularly to reinforce understanding.

By engaging in such exercises, students can develop critical thinking skills and become adept at applying mathematical knowledge to diverse problems, from simple calculations to complex equations.