Problem 93 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 7.2(-2.1)(-2) $$

Step-by-Step Solution

Answer

The result of the operations is 30.24.

Step 1: Multiply First Two Numbers

Start by multiplying the first two numbers: \(7.2\) and \(-2.1\).\[7.2 \times (-2.1) = -15.12\]

Step 2: Multiply the Result with the Third Number

Take the result from step 1 and multiply it by the third number, \(-2\).\[ -15.12 \times (-2) = 30.24 \]

Key Concepts

Multiplication of decimals

When multiplying decimals, the process is very similar to multiplying whole numbers. The main difference is that you have to keep track of the decimal points:

• First, ignore the decimal points and multiply the numbers as if they were whole numbers.
• Count the total number of decimal places from both factors. For instance, if you're multiplying 7.2 and 2.1, you have one decimal place in 7.2 and one in 2.1, so two in total.
• Once you've multiplied the numbers, adjust the product to have the same number of decimal places as the total count you determined earlier.

For example, in the problem where you multiply 7.2 by -2.1, you do:\[7.2 \times 2.1 = 1512 \] (ignoring decimals temporarily). Then, you have two decimal places to consider (one from each number), so the product is \[-15.12\]. Don't forget to account for the sign; since one of the numbers was negative, the result is negative too.

Negative numbers

Handling negative numbers can be tricky at first, but with a few simple rules, it becomes much easier:

• If you multiply a positive number by a negative number, your result is negative.
• If you multiply two negative numbers, the negatives cancel out, resulting in a positive product.
• Always be mindful of the signs, especially in calculations with several numbers.

In the given exercise:- Multiply 7.2 by -2.1 first, which gives a negative result because one factor is negative.- Multiply the subsequent product by another negative (\[-2\]), turning the result positive. Thus, \[-15.12 \times (-2) = 30.24\]. This demonstrates how multiplying two negatives results in a positive number.

Order of operations

The order of operations is a set of rules that determine the sequence in which calculations should be done to ensure consistent results. These rules are often remembered by the acronym PEMDAS:

• Parentheses first
• Exponents (powers and roots, etc.)
• Multiplication and Division (left-to-right)
• Addition and Subtraction (left-to-right)

In our exercise, you only need to focus on multiplication, but these rules make sure calculations are done in the correct order: - Begin by multiplying the first two terms (7.2 and -2.1) as there are no parentheses or exponents to deal with. - Then take the result and multiply by the third number (-2). Recognizing and correctly applying the rules can significantly enhance accuracy in solving any math problem.