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Chapter 2: Problem 12

Multiply the integers. $$(-12)(-2)$$

Short Answer

Expert verified
24

Step by step solution

01

Identify the integers

Identify the two integers that need to be multiplied. The integers are \(-12\) and \(-2\).
02

Multiply the absolute values

Multiply the absolute values of the integers. Absolute value of \(-12\) is 12 and Absolute value of \(-2\) is 2. Therefore, multiply 12 by 2: \[12 \times 2 = 24\]
03

Determine the sign of the product

The product of two negative numbers is always positive. Since both \(-12\) and \(-2\) are negative, their product will be positive. Therefore, \((-12)\times (-2) = 24\)

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Key Concepts

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

Absolute Value
When we talk about the absolute value of a number, we mean its distance from zero on the number line, regardless of direction. In simpler terms, it's the non-negative version of a number. For example, the absolute value of \(-5\) is 5 because that's how far \(-5\) is from zero. Similarly, the absolute value of \(10\) is 10.
To find the absolute value of any integer, we simply ignore the negative sign if it exists. So, the absolute value of \(-12\) turns into 12, and for \(-2\), it becomes 2.
Utilizing absolute values makes it easier to handle multiplication, especially when dealing with negative numbers.
Negative Numbers Multiplication
Multiplying negative numbers can feel tricky, but it's much easier once you understand the rules.
Here is the key point:
  • The product of two negative numbers is always positive.
  • The product of a negative number and a positive number is always negative.

Let's see why these rules hold true with the example provided: \((-12)(-2)\).
First, we multiply their absolute values (ignoring the signs). This gives us \(12 \times 2 = 24\).
Next, because we are multiplying two negative numbers (i.e., -12 and -2), we'll use the rule that says the product of two negative numbers is positive.
Thus, \((-12) \times (-2) = 24\).
Positive Product Rule
The positive product rule is a handy bit of arithmetic that helps us determine the sign of the result when multiplying integers. This rule states that:
  • The product of two negative integers is positive.
  • The product of two positive integers is positive.

Understanding this rule is especially useful for avoiding sign errors in multiplication problems.
In the given exercise, we multiply two negative numbers: \(-12 \times -2\). According to the positive product rule, the result will be positive.
By default, the product of two negative numbers becomes a positive number, as negatives cancel each other out when multiplied.
Remember, knowing and applying this rule makes solving such multiplication problems straightforward and error-free.

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Most popular questions from this chapter

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