Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-8)(-3)$$

Step-by-Step Solution

Answer

The result is \(24\).

Step 1: Understand the rules for multiplying negative numbers

A positive multiplied by a positive results in a positive number, while a negative multiplied by a negative also results in a positive. A negative multiplied by a positive or a positive multiplied by a negative yields a negative result.

Step 2: Apply the rule to given problem

Applying mentioned rule to problem, as both numbers are negative, the result will be positive.

Step 3: Perform the multiplication

Multiply the absolute values of the numbers: \(8*3=24\). Since both original numbers were negative, the result based on rules is positive, thus answer is \(+24\).

Key Concepts

Rules for Multiplication

Understanding multiplication rules, especially involving positive and negative numbers, is essential in mathematics. It helps simplify calculations and predict outcomes accurately. Here’s a simplified version of the rules:

• Multiplying a positive number by another positive number yields a positive result.
• Multiplying a negative number by another negative number also results in a positive number.
• If you multiply a positive number by a negative number or vice versa, the result will be negative.

These rules are crucial as they form the foundation for many calculations involving integers. Identifying the signs of the numbers involved can greatly simplify the process, especially when performing algebraic operations.
As seen in the exercise, multiplying each of the negative numbers, he result is positive.

Absolute Values in Multiplication

The absolute value of a number refers to its magnitude without considering its sign. In mathematical terms, the absolute value of a number is how far it is from zero on the number line, regardless of direction. When dealing with multiplication, understanding absolute values can make calculations easier. Here's why:

• The absolute value of a negative number is the number itself without the negative sign.
• For example, the absolute value of \(-8\) is \(8\), and the absolute value of \(-3\) is \(3\).
• When multiplying numbers, you can first multiply their absolute values to get the magnitude of the result.

So, in our exercise when multiplying \((-8)\) and \((-3)\), you multiply \(8\) and \(3\) to get \(24\). Knowing the rule about signs, you can determine the final sign of the product.

Positive and Negative Results in Multiplication

Deciphering whether the result of a multiplication operation will be positive or negative depends on the signs of the numbers involved. The rules are straightforward:

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

Applying this to the exercise at hand, where both factors \((-8)\) and \((-3)\) are negative, confirms that the product is positive because two negatives make a positive. Thus, understanding these sign rules enables us to predict results accurately without confusion. In solving math problems, mastering these concepts allows for quicker calculations and fewer errors.
This is particularly helpful when dealing with complex algebraic expressions, assuring consistent and accurate results.