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Chapter 1: Problem 723

Use the associative property to simplify. $$ 3(4 x) $$

Short Answer

Expert verified
The simplified expression is \(12x\).

Step by step solution

01

Recognize the Associative Property

The associative property states that the way in which numbers are grouped does not change their product. For multiplication, it can be written as:\[ (a \times b) \times c = a \times (b \times c) \]
02

Apply the Associative Property

Rewrite the expression \(3(4x)\) using the associative property by grouping the constants 3 and 4 first:\[ 3 \times (4 \times x) = (3 \times 4) \times x \]
03

Simplify the Expression

Now, multiply the constants 3 and 4:\[ 3 \times 4 = 12 \]So the expression simplifies to:\[ 12 \times x \]
04

Write the Final Expression

Combine the simplified product with the variable to write the final simplified expression:\[ 12x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elementary Algebra
Elementary algebra is the branch of mathematics that deals with variables, constants, and their relationships. It uses symbols such as letters to represent numbers and arithmetic operations.
Understanding algebra involves working with these symbols to solve equations and simplify expressions.
One important concept is the associative property. This property helps us simplify complex expressions and make calculations easier.
Multiplication
Multiplication is one of the basic operations in mathematics. It involves combining groups of equal sizes.
For example, if you have 3 groups of 4 apples each, you have a total of 12 apples.
This can be written as: \( 3 \times 4 = 12 \).
In the expression given: \( 3(4x) \), multiplication is used to combine the integer 3 with the product of 4 and x.
This process can be simplified using the associative property, where you group numbers together to make the multiplication straightforward.
Simplifying Expressions
Simplifying expressions is a key skill in algebra. It involves rewriting an expression in a simpler form without changing its value.
By simplifying, we make expressions easier to work with and understand.
In the given exercise, we used the associative property to simplify the multiplication expression: \( 3(4x) \).
Here鈥檚 a quick summary of the steps:
  • Recognize the associative property.
  • Group the numbers together.
  • Multiply the constants.
  • Combine the result with the variable.

These steps make the expression \( 3(4x) \) simpler as \( 12x \). Now, the expression is easier to read and work with in further calculations.

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