Problem 5 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-7(-8)$$

Step-by-Step Solution

Answer

The product of \\(-7\\) and \\(-8\\) is \\(56\\).

Step 1: Understanding the problem

We need to find the product of two negative integers, which are \(-7\) and \(-8\). Multiplying two negative integers results in a positive product.

Step 2: Multiplying the absolute values

First, ignore the negative signs and multiply the absolute values of \(-7\) and \(-8\), which are \(7\) and \(8\) respectively. Calculate \(7 imes 8 = 56\).

Step 3: Applying the sign rule for multiplication

Since the rule for multiplying two negative numbers is that the result is positive, the product of \(-7\) and \(-8\) will be positive. So the result of \(7 imes 8\) which is \(56\) remains positive due to the multiplication of two negative numbers.

Key Concepts

Multiplying Negative Numbers

When performing multiplication with negative numbers, it's important to remember the operation's relationship with integers. Negative numbers are simply those that fall below zero on the number line. In multiplication, when you have two negative numbers, such as \(-7\) and \(-8\), the product of these two negative numbers becomes positive. This occurs due to the rules of integer multiplication where - multiplying two negative numbers results in a positive number, - multiplying a positive number by a negative number results in a negative number. To visualize this, think of \(-7\) and \(-8\) as numbers that are repeated movements in the opposite direction on a number line, which results in a positive outcome.

Absolute Value

Understanding absolute value is crucial when dealing with negative numbers and their multiplication. The absolute value of a number is essentially its magnitude or distance from zero, without considering its sign. For example, the absolute value of both \(-7\) and \(7\) is \(7\), and similarly, the absolute value of \(-8\) and \(8\) is \(8\). When we multiply negative numbers, we first find the absolute values of these numbers: - Take \(-7\) : its absolute value is \(7\) - Take \(-8\) : its absolute value is \(8\) We then multiply these absolute values, which in this case gives \(7 \times 8 = 56\). This gives us the magnitude of the answer, and by applying the sign rules later, we can determine whether the result will be positive or negative.

Sign Rules in Multiplication

In integer multiplication, understanding the sign rules can greatly simplify operations. The rules for signs are straightforward:

• If you multiply two numbers with the same sign, whether positive or negative, the result is positive.
• If you multiply two numbers with different signs, the result is negative.

For instance, when multiplying \(-7\) and \(-8\), both numbers share a negative sign. By the sign rules, the result will be positive, despite both being negative individually. Let's apply this practically: - Multiply two positive numbers, \(5 \times 5\) gives \(25\) (positive times positive equals positive)- Multiply a positive and a negative number, \(5 \times -5\) gives \(-25\) (positive times negative equals negative)- Multiply two negative numbers, \(-7 \times -8\) gives \(56\) due to the previous rules. Knowing and applying these rules ensures accurate results when multiplying integers.