Problem 111 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The product of 12 and the difference of 9 and -8

Step-by-Step Solution

Answer

204

Step 1: Identify the operation

The phrase to be translated is 'The product of 12 and the difference of 9 and -8'. The key operations are 'product' and 'difference'.

Step 2: Translate the phrase into an expression

The 'difference of 9 and -8' can be written as 9 - (-8). The 'product of 12 and the difference' means multiplying 12 by this difference.

Step 3: Perform the subtraction

Calculate the difference: 9 - (-8) = 9 + 8 = 17

Step 4: Multiply the results

Now, find the product of 12 and 17: 12 * 17 = 204

Step 5: Write the simplified expression

The simplified expression is 204.

Key Concepts

Numerical Expressions

Numerical expressions are combinations of numbers and operations like addition, subtraction, multiplication, and division. They do not include variables, and their main goal is to simplify or solve the given mathematical phrase using arithmetic operations.

For example, in the exercise, we have the phrase 'The product of 12 and the difference of 9 and -8'. Here, we combined the operations of multiplication (product) and subtraction (difference) to form the numerical expression. The detailed steps show how to break down this phrase and perform the calculations correctly.

• Identify the operations involved
• Translate the phrase into a mathematical expression
• Perform the calculations one by one

Understanding numerical expressions helps develop strong foundational skills in math.

Subtraction

Subtraction is one of the four basic arithmetic operations. It represents taking away one quantity from another. In the context of the exercise, we are dealing with the subtraction involving both positive and negative numbers.

The phrase 'difference of 9 and -8' can be translated into the subtraction expression 9 - (-8). Here's how to handle it:

• Recognize the negative sign next to 8
• Subtracting a negative number is the same as adding its positive counterpart

So, 9 - (-8) becomes 9 + 8. Understanding these rules ensures accuracy in calculations. It transforms the problem to a simpler addition 9 + 8 = 17. Thus, our difference is 17.

Multiplication

Multiplication, another fundamental arithmetic operation, involves combining groups of equal sizes. It's often referred to as repeated addition. In the exercise, after finding the difference, we need to multiply it with another number.

Here's how it works in our example:

• We found the difference to be 17
• The phrase asks for the 'product of 12 and the difference'

This means we multiply 12 by 17: 12 * 17. To perform this multiplication:

• First, multiply 12 by each digit of 17 separately
• Combine these results to get the final product

12 * 17 = 204. Hence, the simplified expression is 204. Mastering multiplication is crucial as it frequently appears in various mathematical contexts.