Problem 109 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 8 added to the product of 4 and \(-10\)

Step-by-Step Solution

Answer

-32

Step 1: Identify operations

The operation in the problem can be represented as follows:3 8 added to (4 times -10)

Step 2: Perform multiplication

The product of 4 and -10 is -40.

Step 3: Perform addition

Now you need to add 8 to the product, so that makes it -40 + 8.

Step 4: Simplify the numerical expression

When you add 8 to -40, you get -32. So the simplified numerical expression is -32.

Key Concepts

Numerical Expression

When we encounter a phrase like '8 added to the product of 4 and \( -10 \)', we are dealing with a numerical expression. This is a mathematical sentence involving numbers and operations but without an equal sign. To convert it into an expression we can break it down into parts. First, identify the operations involved—addition and multiplication in this case. Then arrange the numbers accordingly. Think of this expression as a special code that conveys a sequence of mathematical actions to be performed. It’s crucial to translate this code into mathematical symbols correctly before proceeding with any calculations.

For example, the phrase given is translated to \( 4 \times (-10) + 8 \). Each part corresponds to an action—we multiply 4 by -10 and then add 8 to the result. Numerical expressions like these are the very foundation of algebra and help us build a bridge between real-world problems and mathematical solutions. By learning how to create and interpret these expressions, you open a window into the language of mathematics.

Order of Operations

It's crucial to follow the order of operations when simplifying numerical expressions. This sequence guides us to solve expressions without confusion or error. The general rule often remembered by the acronym PEMDAS or Please Excuse My Dear Aunt Sally stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, even though we don’t have parentheses, we follow PEMDAS by first finding the product of 4 and -10. It's imperative not to rush straight to addition, as mistakenly adding 8 to 4 before multiplying by -10 would result in a completely different and incorrect answer. By adhering to this rule, the simplification process becomes organized and consistent, avoiding potential pitfalls and ensuring accurate results.

Arithmetic Operations

Arithmetic operations are the building blocks for managing numbers and include addition, subtraction, multiplication, and division. Each operation has a specific symbol and a set of rules for how it changes numbers. Multiplication, for instance, combines quantities repeatedly, while addition brings quantities together. These operations are not just randomly applied; they interact in ways that can affect the outcome of a numerical expression significantly.

In our exercise, multiplication (\(4 \times -10\)) was performed first, which resulted in -40, followed by addition (\( -40 + 8\)) to reach the final simplified result of -32. This highlights the precise effects of arithmetic operations on numbers and the importance of performing them in the right sequence.