Problem 5 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$(-3)(7)$$

Step-by-Step Solution

Answer

The result of the multiplication is \(-21\).

Step 1: Identify the numbers

The numbers to be multiplied are: \(-3\) and \(7\)

Step 2: Perform the multiplication

Multiply the absolute value of the numbers: \(3 \times 7 = 21\)

Step 3: Determine the sign

Since one number is positive and the other is negative, our result will be negative. Apply the determined sign to the result from step 2

Step 4: State the final result

The final result of \(-3 \times 7\) is \(-21\).

Key Concepts

Multiplication

Multiplication is one of the basic mathematical operations that we perform in our daily lives. It involves finding the product of two numbers or quantities. Think of multiplication as repeated addition. For example, if you have three groups of seven apples each, adding them repeatedly as \(7 + 7 + 7\) will give you the same result as multiplying them: \(3 \times 7\). When multiplying numbers, each number is referred to as a "factor," and the resulting number is called the "product."

• Here, \(-3\) and \(7\) are factors.
• The process involves multiplying their absolute values first.

Understanding this concept forms the foundation for dealing with more complex mathematical problems.

Negative Numbers

Negative numbers are numbers less than zero. Understanding how they behave during mathematical operations is crucial. When multiplying negative numbers, the sign of the result depends on the number and signs of the factors involved.

• If both numbers are negative, the product is positive.
• If one number is positive and one is negative, the product is negative.
• A negative multiplied by a positive number decreases the value, as seen in \(-3 \times 7 = -21\).

This showcases the rule that a negative multiplied by a positive gives a negative outcome. Always consider signs to ensure you are applying the multiplication rules correctly.

Mathematical Operations

Mathematical operations serve as the building blocks for more complicated equations and problems. The four basic operations are addition, subtraction, multiplication, and division. Each operation has its unique set of rules and properties.
In the operation example given, multiplication and the sign rule for negative numbers are applied.

• Multiplication is performed first by determining the absolute values: \(3\) and \(7\).
• The sign of the product is then determined by multiplying a negative number by a positive one, resulting in \(-21\).

Keeping operations organized and following their properties ensures accuracy in problem-solving. Mastery of these operations and their rules enables you to tackle more advanced mathematics with confidence.