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

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-9)(-5)$$

Step-by-Step Solution

Answer

The result of \((-9) \times (-5)\) is \(45\).

Step 1: Multiply the numbers

Simply multiply the two numbers together, without considering the negative signs. In this case, \(9 \times 5 = 45\).

Step 2: Apply the rule of signs

In multiplication, the rule of signs says that when you multiply two negative numbers, the result is positive. Thus, the answer to \((-9) \times (-5)\) is positive.

Key Concepts

Rule of Signs

Understanding the rule of signs is essential when dealing with multiplication involving negative numbers. This straightforward principle dictates the sign of the product based on the signs of the factors involved.

When multiplying two negative numbers, such as in the exercise \( (-9)(-5) \), the rule of signs tells us that the product will be positive. This might be surprising initially, but it makes sense if we consider what negative numbers represent. A negative number can be seen as the opposite of a positive number. So, multiplying two 'opposites' yields a 'positive' result, just as two wrongs make a 'right' in common parlance.

The rule of signs also covers other scenarios, such as:

• Multiplying a positive number by a negative number results in a negative number.
• Multiplying two positive numbers, as always, yields a positive number.

Remembering this rule can simplify the process of working through algebra problems and avoid common mistakes.

Integer Multiplication

Multiplication of integers might seem daunting at first, but once the rules are understood, it's quite straightforward. In the exercise, we multiply two integers, \( -9 \) and \( -5 \). The very first step, as shown in the solution, is to multiply the absolute values of the integers, which are simply the numbers without their signs.

In this example, we multiply 9 by 5 to get 45. This step is crucial because it separates the complexity of dealing with signs from the act of multiplication itself. Once the multiplication part is completed, we then apply the rule of signs to determine the final sign of the product based on the signs of the original integers.

Multiplying integers is a fundamental skill in mathematics that is applied constantly in algebra and higher math. Grasping this concept builds a solid foundation for solving more complex problems, including those in algebraic expressions.

Algebraic Expressions

Algebraic expressions are a bedrock of algebra that contains numbers, variables, and operators (such as addition, subtraction, multiplication, and division). Understanding how to handle the multiplication of negative numbers within these expressions is crucial.

While the example \( (-9)(-5) \) does not contain variables, the same rules apply when they do. Multiplying numbers within algebraic expressions involves the same steps: multiply the absolute values first, then apply the rule of signs. For example, if you have the expression \( (-x)(-y) \) where \( x \) and \( y \) are positive numbers, the result would be a positive expression (\( xy \) because 'a negative times a negative equals a positive'.

It's vital to handle the coefficients and variables carefully while simplifying or expanding algebraic expressions. Building an intuition for multiplication rules and signs helps avoid errors and leads to a smoother experience with algebra as a whole.