Problem 19 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$3(-1.2)$$

Step-by-Step Solution

Answer

-3.6

Step 1: Identify the numbers to multiply

The numbers to multiply are 3 and -1.2.

Step 2: Perform the multiplication

Multiplication is the arithmetic operation of scaling one number by another. Therefore, simply multiply the two numbers: 3 multiplied by -1.2 equals -3.6.

Key Concepts

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics, including addition, subtraction, multiplication, and division. These operations form the basis for much more complex mathematical concepts, but mastering them is crucial in itself. When it comes to multiplication, it's helpful to think of it as repeated addition. For instance, multiplying 3 by -1.2 is akin to adding -1.2 to itself three times.

Multiplication can also be viewed as scaling; you're essentially scaling the number -1.2 by a factor of 3. This operation is symmetric, which means that the order in which you multiply the numbers does not affect the result: multiplying 3 by -1.2 yields the same result as multiplying -1.2 by 3. The key is to apply the multiplication algorithm properly, which involves multiplying the absolute values of the numbers and then determining the sign of the result based on the rules of sign multiplication.

Negative Number Multiplication

Multiplying negative numbers might initially seem a bit tricky, but once you understand the rule, it becomes straightforward. Here's the essential principle: the product of two numbers with the same sign is positive, and the product of two numbers with different signs is negative.

So, when multiplying a positive number by a negative number, as in our example with 3 and -1.2, the result will always be negative. It's helpful to first multiply the absolute values—ignoring the signs—to find that 3 times 1.2 equals 3.6. Then, apply the rule for signs to understand that since the factors have different signs, the product, in this case, will be negative: \( -3.6 \). Remembering this rule will ensure success with multiplication of real numbers, whether they're positive or negative.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the language through which we translate mathematical problems into a form that can be analyzed and solved. In the context of multiplication, algebraic expressions can include terms that need to be multiplied together.

When dealing with multiplication in algebraic expressions, it's important to consider the order of operations, usually abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Since multiplication comes before addition and subtraction in this hierarchy, it must be performed first when simplifying expressions. For our specific exercise, multiplication is the central operation, so we would prioritize it over any other potential operations within an algebraic expression.