Problem 15 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (-4)(-8) $$

Step-by-Step Solution

Answer

Answer: 32

Step 1: Rewrite the expression

First, rewrite the expression to make it more understandable: $$ (-4)(-8) = -4 \times -8 $$

Step 2: Apply rules for multiplying negative numbers

The product of two negative numbers is a positive number. So when multiplying -4 by -8, we will first multiply the numbers themselves, then consider the result to be positive: $$ -4 \times -8 = 4 \times 8 $$

Step 3: Multiply the numbers

Finally, multiply 4 by 8 to get the result: $$ 4 \times 8 = 32 $$ So, the value of the expression \((-4)(-8)\) is 32.

Key Concepts

Understanding Algebraic Expressions

At the heart of algebra lies the concept of an algebraic expression, which is a combination of numbers, variables, and arithmetic operations. To comprehend algebraic expressions, one must be familiar with the components that make them up and the operations that can be performed on them.

An expression like \( (-4)(-8) \) might appear intricate at first, but it's simply an algebraic expression involving numbers and multiplication. When approaching such expressions, the initial step is to identify the numbers and operations involved. In our example, we have two numbers, \( -4 \) and \( -8 \) which are to be multiplied together.

Rules of Multiplication

Students often learn early on that multiplication is a shortcut for repeated addition. However, multiplication rules become slightly more complex when we introduce negative numbers. To multiply numbers effectively, always remember the following key rules of multiplication:

• If both numbers are positive, the product is positive.
• If one number is positive and the other is negative, the product is negative.
• If both numbers are negative, the product is positive.

Our exercise shows the multiplication of two negative numbers, where these rules are applied to determine the sign of the product.

Arithmetic Operations

Arithmetic operations are the foundation of basic mathematics, including addition, subtraction, multiplication, and division. These operations are the building blocks for solving mathematical problems and understanding more complex mathematical concepts.

In our given expression, the arithmetic operation involved is multiplication. Mastering how to perform arithmetic operations with both positive and negative numbers is crucial for solving algebraic expressions efficiently.

Negative Number Multiplication

Multiplying negative numbers can seem counterintuitive at first, but once the underlying rule is understood, it becomes straightforward. The rule, as seen in the exercise, is that the product of two negative numbers is a positive number. This aspect of negative number multiplication is critical as it applies universally, whether one is dealing with simple numbers, variables, or more complex expressions.

By remembering that a negative times a negative equals a positive, as demonstrated in the solution \( -4 \times -8 = 4 \times 8 = 32 \), students can approach these types of problems with confidence. Introducing this concept with clear examples, like our exercise, reinforces students' understanding and application of the rule.