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Chapter 1: Problem 13

Perform the following operations with real numbers. $$ (-9)(-12) $$

Short Answer

Expert verified
The result is 108.

Step by step solution

01

Understanding the Problem

The problem is asking us to multiply two negative real numbers: \(-9\) and \(-12\).
02

Applying the Rule for Multiplying Negative Numbers

When you multiply two negative numbers, the result is a positive number. Therefore, the product of \(-9\) and \(-12\) will be positive.
03

Calculating the Product

Multiply the absolute values of the numbers. Ignore the negative signs for this step, so calculate 9 times 12.
04

Final Calculation

Calculate \(9 \times 12 = 108\). Because we are multiplying two negatives, the result is positive. Therefore, \((-9)(-12) = 108\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplying Negative Numbers
When you multiply two negative numbers, the product is positive. This may seem a bit counterintuitive since dealing with negative numbers can feel tricky. However, think of it this way: a negative sign can be viewed as indicating the opposite direction.

* Multiplying two negatives is like reversing direction twice, which brings you back to moving in a positive direction.
* For example, in simple terms, if you "owe" (negative) something twice, you can "get" (positive) in return.

Therefore, in the expression \(-9 \times -12\), the negative signs "cancel out," resulting in a positive outcome. Remember this rule: a negative times a negative always equals a positive!
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It's always non-negative.

* For example, the absolute values of \-9\ and \-12\ are 9 and 12, respectively.
* When multiplying negative numbers, calculating their absolute values first can simplify the operation. To solve \(-9 \times -12\), first consider their absolute values, which turns the multiplication into 9 \times 12.

This step ensures that you are only focusing on the quantity without worrying about the signs at this stage.
Positive Product
The result of multiplying two numbers is called their product. A positive product means the result is greater than zero.

* When multiplying two negative numbers, we've learned that the outcome is a positive product. This is because multiplying those negative values gives us positive results.

In our example, \(-9 \times -12 = 108\), the final answer is a positive product. The product of 9 \times 12 is 108, and because we're multiplying two negative numbers, the final answer remains 108鈥攂y following the rule that two negatives make a positive. Remember, achieving a positive product from two negatives isn't just a trick; it's a foundational rule of arithmetic.

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