Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Use an associative property to rewrite each algebraic expression. Once the grouping has been changed, simplify the resulting algebraic expression. $$7(4 x)$$

Step-by-Step Solution

Answer

The simplified expression is \(28x\).

Step 1: Apply associative property

The associative property allows to change the grouping of the factors. This means that the expression \(7(4x)\) can be rewritten as \((7\times 4)x\).

Step 2: Simplify the expression

Multiplying 7 by 4 gives 28. The expression now becomes \(28x\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operations. They can include addition, subtraction, multiplication, and division. For example, in the expression \(7(4x)\), 7 and 4 are numbers, and \(x\) is a variable. Variables represent unknown values and can change depending on the situation. Writing expressions helps to generalize mathematical situations and find solutions to problems where specific values are unknown.

Algebraic expressions can be simple, like \(x + 2\), or more complex, with multiple terms, like \(3x^2 - 2x + 7\). When working with expressions, understanding how to manipulate them using properties of arithmetic is crucial, which leads us to the associative property.

Simplification

Simplification is the process of reducing an algebraic expression into its simplest form while retaining its original value. This involves combining like terms and performing basic arithmetic operations. Simplification makes expressions easier to interpret and solve.

The process of simplification might involve several steps, such as:

• Rewriting expressions using properties like the associative and distributive properties.
• Combining like terms, which are terms that involve the same variables raised to the same powers.
• Performing arithmetic operations on constants and coefficients.

In our example, the expression \(7(4x)\) was simplified to \(28x\) by first changing the grouping of terms and then performing the multiplication. This final expression is easier to understand and work with, especially when solving equations.

Multiplication in Algebra

Multiplication in algebra involves combining variables and constants while respecting the rules of arithmetic. In algebra, multiplication is often implicit, especially when variables are involved. For instance, in \(4x\), it is understood as \(4\times x\).

When multiplying, it's essential to:

• Acknowledge the use of properties, like the associative property, which influences the grouping of factors. This means you can multiply numbers in any order and still get the same result.
• Combine coefficients (numbers in front of variables) and apply multiplication throughout the expression.
• Be mindful that multiplication of like bases (variables) involves adding their exponents, though it didn't apply to our original example \(7(4x)\).

In our exercise, the expression was rewritten as \((7\times 4)x\) using the associative property, which was then simplified to \(28x\). Understanding multiplication in algebra is crucial to successfully manipulating and solving algebraic expressions.