Problem 15 - Beginning and Intermediate Algebra

Question

Find each product. \(-10(-12)\)

Step-by-Step Solution

Answer

120

Step 1: Understand the Problem

The task is to find the product of two numbers: -10 and -12. Note that both numbers are negative.

Step 2: Multiply the Numbers

Multiply the absolute values of the two numbers first: \(10 \times 12 = 120\)

Step 3: Apply Sign Rules

According to the multiplication rules for integers, the product of two negative numbers is positive. Hence, the product of -10 and -12 will be positive. So, \(-10 \times -12 = 120\)

Step 4: Conclusion

By combining the absolute values and applying the sign rule, the final answer is 120.

Key Concepts

Integer Multiplication

To start understanding how to multiply integers, remember that an integer is any whole number, including negative numbers, positive numbers, and zero. When it comes to multiplication, the process is simple: we multiply the absolute values of the numbers and then apply the appropriate sign based on the rules of multiplication for integers.

It helps to break the process down into steps:

• First, ignore the signs and multiply the numbers as if they were both positive. This gives you the 'magnitude' of the product.
• Second, apply the sign rules for multiplication to determine whether the final product should be positive or negative.

In our example from the exercise, \( -10(-12) \), we start by multiplying the absolute values: \( 10 \times 12 = 120 \). The result here, 120, represents the magnitude of our product.

Absolute Value

Understanding the absolute value is crucial for integer multiplication. The absolute value of a number is essentially its distance from zero on the number line, regardless of its direction. In other words, the absolute value of a number is always non-negative.

For example:

• The absolute value of -10 is 10.
• The absolute value of -12 is 12.

In our example: \( -10(-12) \), you first consider the absolute values of -10 and -12, which are 10 and 12, respectively. You then multiply these absolute values together, getting \( 10 \times 12 = 120 \).

Once we have this 'magnitude' of the product, we must apply the sign appropriately, which brings us to the next important topic.

Sign Rules for Multiplication

The sign rules for multiplication play a key role in determining the final sign of our product when dealing with integers. These rules are straightforward and crucial to understand:

For multiplication:

• If both numbers are positive, the product is positive.
• If one number is positive and the other is negative, the product is negative.
• If both numbers are negative, the product is positive.

Applying this to our example \( -10(-12) \):

Since both -10 and -12 are negative numbers, according to our sign rules, the product of two negative numbers is a positive number.

So, after calculating the magnitude \( 10 \times 12 = 120 \), we apply the sign rule to conclude that \( -10 \times -12 = 120 \). Therefore, the final answer is positive 120.