Problem 34 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (-8)(7) $$

Step-by-Step Solution

Answer

Answer: The product of -8 and 7 is -56.

Step 1: Multiply the integers without their signs

Let's first multiply the numbers as if they were both positive. It will look like this: $$ 8 * 7 = 56 $$ So, the multiplication of their absolute values results in 56.

Step 2: Determine the sign of the result

Now, let's consider the signs of the original, given numbers. One of the numbers (-8) is negative, and the other number (7) is positive. When multiplying a negative number by a positive number, the result will always be negative. Therefore, our final result will be negative.

Step 3: Combine the result and sign

Now, we know that the product of the absolute values of the two given numbers is 56, and the sign of the result is negative. To obtain the final result, we simply add the negative sign to the number 56: $$ (-1) \cdot 56 = -56 $$ So, the value of the expression is -56.

Key Concepts

Understanding Absolute Value

Absolute value is a concept that represents the distance of a number from zero on a number line. It is always expressed as a positive number, regardless of whether the original number is positive or negative. For example, the absolute value of both \(-8\) and \(8\) is \(8\). This is because both are exactly 8 units away from zero.

Absolute value is denoted by two vertical bars around the number, like this: \(|-8| = 8\) and \(|8| = 8\). It helps us understand the magnitude of a number without considering its direction or sign. This concept is especially useful in integer multiplication, where you initially ignore the signs to multiply the base values. Once you have the result of the absolute values, you can then apply the correct sign based on the rules for multiplying integers.

Exploring Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign in front of them (e.g., \(-8\)). They are located to the left of zero on the number line. The further left you go, the smaller the value becomes.

When dealing with multiplication, remember:

• Multiplying two negative numbers results in a positive number. This is because the negatives cancel each other out.
• Multiplying a negative number by a positive number results in a negative number, as the sign of the negative number dominates.

In our exercise, \(-8\) is negative. When multiplied by \(7\), a positive number, the result is \(-56\). This maintains the rule that a negative times a positive is negative.

Knowing Positive Numbers

Positive numbers are numbers greater than zero and do not have a sign in front of them (e.g., \(7\)). They are found to the right of zero on the number line. Positive numbers indicate quantities that increase or positive balances.

In multiplication, positive numbers follow straightforward rules:

• Multiplying two positive numbers results in a positive product.
• When multiplying with a negative number, the result is dictated by the negative number's sign.

In the given expression \((-8)(7)\), 7 is positive. Despite this, because of the presence of a negative factor (\(-8\)), the product remains negative. Understanding this dynamic is critical in mathematical operations.