Problem 79 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (-2)(-2)(-2)(-2) $$

Step-by-Step Solution

Answer

The answer to the operation is 16.

Step 1: Understand the Operation

The given operation is the multiplication of four negative numbers, which are all -2.

Step 2: Simplify in Pairs

First, we will multiply the first two numbers: \[ (-2) \times (-2) = 4 \]Since the multiplication of two negative numbers results in a positive number.

Step 3: Multiply the Next Pair

Next, multiply the remaining pair of numbers: \[ (-2) \times (-2) = 4 \]Again, the result is positive 4.

Step 4: Multiply the Results

Now, multiply the two results obtained from steps 2 and 3: \[ 4 \times 4 = 16 \]

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They're typically represented with a minus sign in front, like -1, -2, or -3. Negative numbers are used to describe values below a defined point, such as temperature below freezing or altitude below sea level. Understanding negative numbers is important because they behave differently in arithmetic operations compared to positive numbers. For instance, adding two negative numbers results in a larger negative number. If you add -3 and -5, you get -8. When multiplying, however, the rules are different, especially when negative numbers interact with positive ones. We'll explore how these rules apply in the following sections.

Multiplication Steps

Multiplication is one of the four basic operations in arithmetic and involves finding the total number of objects in a certain number of groups. When dealing with negative numbers, it's essential to follow specific steps to ensure accurate results.

• Identify each negative and positive number: Start by clearly noting which numbers in your operation are negative and which are positive.
• Multiply in pairs: Begin by multiplying two numbers at a time. For example, multiplying two negative numbers results in a positive number.
• Combine results: After obtaining results from the pairwise multiplications, you multiply these results together to get the final product.

For instance, in calculating \( (-2) \times (-2) \times (-2) \times (-2) \), you multiply the first two negatives to get a positive, and then repeat for the next pair. Finally, you multiply the two resulting positives. Following these steps allows for orderly calculation and helps in avoiding mistakes.

Positive and Negative Product

The product of a series of numbers depends heavily on whether the numbers are positive or negative. Here's a simple rule to remember:

• Multiplying two negative numbers gives a positive product.
• Multiplying a negative and a positive gives a negative product.
• If you multiply an even number of negative numbers, the product is positive. If it's an odd number of negative numbers, the product is negative.

In the operation \((-2) \times (-2) \times (-2) \times (-2)\), we multiply four negative numbers. Pairwise, each multiplication flips the sign: the first negative times the second results in a positive, and so forth. Eventually, since all numbers have been paired into negatives, we end up with a positive product because four is an even number.Remembering these multiplication rules helps simplify complex problems and ensures accurate results when dealing with integers.