Problem 17 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-4(3)(-2)$$

Step-by-Step Solution

Answer

24

Step 1: Multiply the first two numbers

Start by multiplying the first two numbers in the expression: \(-4 \times 3 = -12\)The new expression becomes \(-12 \times (-2)\).

Step 2: Multiply the result by the last number

Now, multiply the result from Step 1 by the last number:\(-12 \times (-2) = 24\) Remember that multiplying two negative numbers results in a positive number.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. These numbers are represented with a minus sign \((-\)). In mathematics, dealing with negative numbers can seem tricky at first, but with some simple rules, it becomes quite clear.
Negative numbers arise in many different real-world contexts, like representing debts, temperatures below freezing, or elevations below sea level. When you multiply two negative numbers, the result is a positive number.

• The product of two negative numbers is a positive number. For instance, \(-2 \times -3 = 6\).
• The product of a positive number and a negative number is a negative number. For example, \(4 \times -5 = -20\).

Understanding these rules aids in solving complex mathematics questions quickly and accurately, just like the exercise shown.

Order of Operations

When working with mathematical expressions, it's crucial to perform operations in the right order to get the correct answer. This concept is known as the "order of operations." There is a simple acronym to help remember the order: PEMDAS.
- **Parentheses** - **Exponents** - **Multiplication** - **Division** - **Addition** - **Subtraction** Multiplication and division should be performed from left to right, as they appear in the expression, and the same rule applies for addition and subtraction.
In the original exercise, the multiplication was performed in the order it appeared, which led to the correct result. Remembering to follow the order of operations and keeping it consistent every time allows for avoiding any miscalculations.

Integer Operations

Integers are whole numbers that can be either positive or negative, including zero. They are fundamental in mathematics because they allow us to perform all sorts of operations, like addition, subtraction, multiplication, and division.
When dealing with integer operations, it's important to follow some basic rules:

• Adding integers: Adding a larger positive to a smaller negative number results in a positive number. For example, \(6 + (-3) = 3\).
• Subtracting integers: Subtracting a larger positive number from a smaller positive number results in a negative number. Example: \(5 - 7 = -2\).
• Multiplying integers: As discussed, multiplying two negative integers gives a positive result. Multiplying a positive with a negative integer results in a negative outcome.
• Dividing integers: Dividing follows the same rules as multiplication. A positive divided by a negative gives a negative result, and two negatives divided yield a positive.

These guidelines help to ensure that operations with integers are consistent and accurate, as demonstrated in the multiplication exercise between the integers \(-4\), \(3\), and \(-2\).