Problem 71 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. six times the product of 4 and a number

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is 24\(x\).

Step 1: Identify the Numbers

The phrase mentions three numbers: six (6), four (4), and an unknown number, which is referred to as \(x\) in the given problem.

Step 2: Convert into Algebraic Expression

The English phrase 'six times the product of 4 and a number' translates into the expression: 6 * (4 * \(x\))

Step 3: Simplify the Expression

The multiplication operation is associative, which means the way in which numbers are grouped does not change their product. Simplify 6 * (4 * \(x\)) into 24 * \(x\) or 24\(x\).

Key Concepts

Simplification

Simplification in algebra is the process of making an expression easier to read and work with. It involves reducing the expression to its simplest form while retaining the equality. For example, when we have the expression 6 times the product of 4 and a number, we can simplify it for ease of calculation and understanding.
Starting with the complex expression 6 * (4 * \(x\)), we can streamline this by calculating the product inside the parentheses first. Compute 4 * \(x\), which gives us 4\(x\). Next, multiply this result by 6, leading to a new simplified expression, 24\(x\).
By simplifying, we not only make calculations more straightforward but also enhance clarity when communicating mathematical ideas, which is critical in solving more complex algebraic problems.

Multiplication

Multiplication in algebra is an operation that combines numbers (or variables) to reach a product. It’s one of the core operations in arithmetic and is vital in expressing relationships between numbers.
When dealing with algebraic expressions, multiplication often involves variables. In our case, we initially interpret the phrase as multiplying 4 by a number \(x\), giving us 4\(x\) as this intermediate result. The next step of the operation is to multiply 4\(x\) by 6.
This step leads to multiplying 6 by the entity 4\(x\). It's important to follow the sequence of operations meticulously. The multiplication can be expanded to 6 * 4 * \(x\) which simplifies into 24\(x\). This process highlights the bond between numbers and variables, fundamental in expressing real-world quantities in algebra.

Associative Property

The associative property is a fundamental rule in arithmetic and algebra. It refers to how numbers are grouped in an operation and assures us that changing the grouping does not affect the result of the multiplication or addition operations. In simpler terms, regardless of how the numbers are bracketed, the outcome remains the same.
For multiplication, the associative property is critical. For example, the expression 6 * (4 * \(x\)) can be rearranged as (6 * 4) * \(x\) due to the associative property. This grouping flexibility means we calculate the result step-by-step in an order that simplifies the equation without altering the result.
No matter how you choose to group the numbers in multiplication, you will still end up with the correct conclusion. Understanding this property helps you manipulate and simplify expressions effectively, making algebraic problems less intimidating.