Problem 21 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \text { (3) }(-12) $$

Step-by-Step Solution

Answer

Answer: The value of the expression (3)(-12) is -36.

Step 1: Identify the problem

In this step, we identify that we have an expression that is composed of a multiplication operation between two numbers: 3 and -12.

Step 2: Perform the multiplication

Now, we need to multiply 3 and -12 in order to find the value of the given expression. We can either use the Commutative Property and multiply -12 by 3 or multiply 3 by -12. So, (3)(-12) = 3 × -12 = -36 Thus, the value of the expression (3)(-12) is -36.

Key Concepts

Arithmetic Operations

Arithmetic operations are the foundation of mathematics. They include addition, subtraction, multiplication, and division. In the context of our exercise, we focus on multiplication, which combines groups of equal size to find the total amount.

Understanding multiplication with integers (which are whole numbers that can be positive, negative, or zero) requires us to observe the signs as well as the numbers. The basic rule to remember is: multiplying two positive numbers or two negative numbers always gives a positive result, while multiplying a positive number with a negative one results in a negative outcome.

In the exercise \( (3)(-12) \), we are dealing with multiplying a positive integer by a negative one. This means our result will be negative. The magnitude of the product is found by multiplying the absolute values of the two numbers, which gives us \( 3 \times 12 = 36 \). Since the multiplier, \( -12 \), is negative, our final answer is \( -36 \).

Commutative Property

The commutative property of multiplication states that changing the order of the numbers we are multiplying doesn't change the product. In simpler terms, \( a \times b = b \times a \). This property is a cornerstone in arithmetic as it allows for flexibility in calculations and is essential for understanding the multiplication of integers.

Let's apply this property to our exercise. It doesn’t matter if we multiply \( 3 \times -12 \) or \( -12 \times 3 \), the result is the same: \( -36 \). The commutative property ensures that the sequence in which the integers appear can be switched, granting us confidence that the product remains consistent, which is particularly helpful when solving complex problems or when estimating products mentally.

Negative Numbers

Negative numbers are numbers less than zero. They are usually represented with a minus sign (-) in front. In the temperature scale, for example, below zero temperatures are marked as negative. In the context of multiplication, knowing the behavior of negative numbers is crucial.

A negative number multiplied by a positive number gives a negative product, as seen in our exercise. However, if both numbers in a multiplication operation are negative, the result is a positive product. Why is that? It's like saying 'a negative times a negative equals a positive,' which stems from the idea that a negative number is an opposite. So, performing an opposite action twice brings us back to our starting point, a positive result.

Understanding the rules for multiplying negative numbers helps avoid mistakes and allows us to confidently tackle more complex arithmetic involving integers.