Problem 16 - Prealgebra and Introductory Algebra

Question

Multiply. $$(-3)(-6)$$

Step-by-Step Solution

Answer

The result is 18.

Step 1: Identify the numbers to be multiplied

Here, the numbers are -3 and -6.

Step 2: Application of rule of signs

According to the rule of signs in multiplication, the product of two negative numbers is always positive.

Step 3: Multiplication

Multiply the absolute values of the numbers without considering the signs. We get \(3 \times 6 = 18\). Then, apply the sign rule. The answer would be positive, so, it's 18.

Key Concepts

Rule of Signs

The rule of signs is a fundamental principle in mathematics that helps us determine the sign of a product when multiplying integers. This rule is particularly useful when dealing with both positive and negative numbers. Here's how the rule works:

• When two positive numbers are multiplied, the result is always positive.
• When two negative numbers are multiplied, the result is also positive. This happens because the negative signs "cancel out."
• When a positive number is multiplied by a negative number (or vice versa), the result is negative. This is because the negative sign influences the entire product.

In the exercise provided, both numbers are negative (-3 and -6), so according to the rule of signs, their product will be positive. This might seem counter-intuitive at first but remembering this simple rule can clear up confusion.

Absolute Value Multiplication

When we multiply integers, it's often easier to first consider their absolute values. The absolute value of a number is its distance from zero, disregarding whether it's positive or negative. It is denoted by enclosing the number in vertical bars, for example, \|-3\| = 3.

By focusing on absolute values first, we simplify the multiplication process.

• For the numbers -3 and -6, their absolute values are 3 and 6.
• We multiply these absolute values: \(3 \times 6 = 18\).

After calculating the product of the absolute values, we can then apply the rule of signs to determine the final sign of the product, which in this case remains positive.

Negative Numbers Multiplication

Multiplying negative numbers can seem tricky, but it becomes straightforward once we grasp a few basic concepts. Let's break it down:

When you multiply two negative numbers, each of these contributes to "flipping" the result to a positive. Here's a step-by-step breakdown of what's happening:

• First, ignore the negative signs and multiply their absolute values, just as you would with positive integers.
• Then, remember to apply the rule of signs: since both numbers are negative, the negatives cancel out, and the result is positive.

This is why in the example of multiplying -3 and -6, we consider \((-3)(-6)\) to become \(18\), as the canceling negatives turn the product positive. Once you're familiar with this process, it becomes second nature and makes working with negative numbers much easier.