Problem 71 - Introductory and Intermediate Algebra for College Students 4th

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 phrase 'six times the product of 4 and a number' translates to the algebraic expression \( 6 \times (4 \times x) \) and simplifies to \( 24x \)

Step 1: Identifying the Operations and Associated Numbers

The phrase 'six times the product of 4 and a number' means that we should multiply 6, 4 and \( x \) together. The phrase uses 'times' and 'product' to signal that these three values should be multiplied.

Step 2: Translating into an Algebra Expression

The transition into a mathematical expression becomes, in order: 'six (6) times (multiplied by) the product (the result of multiplication) of 4 and a number (\( x \))' translates to \( 6 \times (4 \times x) \).

Step 3: Simplifying the Algebra Expression

The expression can be simplified as the multiplication operation is associative - it does not matter how the numbers are grouped. This means 6 multiplied by 4 multiplied by \( x \) can simply be written as \( 24x \)

Key Concepts

Translating Phrases to Algebra

Understanding how to convert English phrases into algebraic expressions is a key skill in algebra. It translates word problems into a form that you can solve using mathematical operations. Let's break down the phrase from the exercise: "six times the product of 4 and a number." Here, "product" signals multiplication. When you see "product of 4 and a number," you should think about multiplying, meaning you take 4 and multiply it by the unknown, which is represented as \( x \). The phrase "six times" indicates that we multiply "six" by whatever follows it, which is the whole "product of 4 and a number". Thus, we translate this whole phrase into the algebraic expression \( 6 \times (4 \times x) \). This method involves identifying keywords like "times" and "product" and knowing the operations they represent.

Simplifying Expressions

Simplifying an algebraic expression makes it easier to handle and understand. Once you have an expression like \( 6 \times (4 \times x) \), you can simplify it by performing the multiplications step by step.

• First, calculate the multiplication inside the parentheses: \( 4 \times x \). This operation simply remains as \( 4x \) because you cannot combine these terms unless \( x \) has a specific value.
• Next, multiply the result by the number outside the parentheses: \( 6 \times 4x \).
• This results in \( 24x \), because you multiply the numerical coefficients (6 and 4) together.

Simplifying expressions means performing operations, such as multiplication, on numbers and combining like terms as much as possible. The result, \( 24x \), is easier to use in further calculations or in a broader algebra problem.

Multiplication in Algebra

Understanding multiplication in algebra involves recognizing how numbers and variables interact within expressions. The associative property of multiplication states that how you group numbers does not affect the product. In the translated expression \( 6 \times (4 \times x) \), we apply multiplication twice.

• The first multiplication is \( 4 \times x \), which keeps 4 as a coefficient of \( x \) in the term \( 4x \).
• The second is \( 6 \times (4x) \), illustrating how a number multiplies a term to scale both the coefficient and the variable part.

This flexibility allows us to rewrite expressions in simpler forms. Recognizing and using the associative property consolidates operations in complex algebraic expressions, making it easier to solve equations and understand other algebraic manipulations.