Problem 34 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ (-4)(-6) $$

Step-by-Step Solution

Answer

24

Step 1 - Identify the numbers

The given numbers to multiply are -4 and -6.

Step 2 - Understand the multiplication rule for negative numbers

When two negative numbers are multiplied, the product is positive.

Step 3 - Multiply the absolute values

Multiply the absolute values of the numbers: \[ | -4 | \times | -6 | = 4 \times 6 = 24 \]

Step 4 - Apply the sign rule

Since both numbers are negative, their product is positive. Therefore, \[ (-4) \times (-6) = +24 \]

Key Concepts

absolute value

When learning about the multiplication of negative numbers, understanding the concept of absolute value is crucial.
The absolute value of a number is its distance from zero on the number line, regardless of direction.
For example, the absolute value of both -4 and 4 is 4, denoted as \[ | -4 | = 4 \].
In the original exercise, we multiply the absolute values of -4 and -6. Here it means taking simply 4 (from -4) and 6 (from -6).
This process removes the negative signs when performing the arithmetic operation, making the calculations easier to handle.

sign rules for multiplication

Sign rules are guidelines for determining the sign of a product when multiplying numbers.
These rules are straightforward and very important in mathematics.
Specifically:

• If you multiply two positive numbers, the product is positive.
• If you multiply a positive number and a negative number, the product is negative.
• If you multiply two negative numbers, the product is positive.

In the given problem (-4) * (-6), we're dealing with two negative numbers.
According to the sign rules, multiplying these will yield a positive product.

positive product of two negatives

It might seem surprising that multiplying two negative numbers results in a positive product.
However, this is because of the inherent properties of negative and positive numbers.
When you multiply two negative numbers, you're essentially multiplying their absolute values and then determining the sign using the sign rules.
Therefore, when we multiply -4 by -6, we find that \[ 4 \times 6 = 24 \], and since both numbers are negative, the product is positive.
So, the final answer is \[(-4) \times (-6) = +24 \], as we've seen in the exercise.