Problem 13 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(8+4(3 x+6)\)

Step-by-Step Solution

Answer

32 + 12x

Step 1: Distribute Inside the Parentheses

First, distribute the 4 to both terms inside the parentheses: 8 + 4(3x) + 4(6)

Step 2: Simplify the Distributed Terms

Now, simplify the expressions from the distribution step: 8 + 12x + 24

Step 3: Combine Like Terms

Combine the constant terms (8 and 24): 32 + 12x

Key Concepts

Distribution in Algebra

In algebra, distribution is a crucial step when working with expressions that include parentheses. Imagine you have an expression like this:
o 8 + 4(3x + 6).
The first step is to remove the parentheses by distributing the multiplication over addition within the parentheses. Distribution means multiplying each term inside the parentheses by the number outside. In our example, you distribute the 4 to both 3x and 6 like this:
8 + 4(3x) + 4(6),
which simplifies to
8 + 12x + 24.
Distribution removes the parentheses and gives us equally weighted terms in an algebraic expression. This process is sometimes called the distributive property.

Combining Like Terms

Next, you need to combine like terms to further simplify the expression. Like terms are terms that contain the same variables raised to the same power. In our example, after distributing, we have:
8 + 12x + 24.
Here, 8 and 24 are constants (they don't have variables attached), and 12x is a term with a variable. Combining like terms essentially means adding or subtracting these constants together. So we take 8 and 24 and combine them:
8 + 24 = 32.
So, the expression now looks like this:
32 + 12x.
This step ensures all constants are collapsed into a single term and any terms with variables are simplified as much as they can be.

Simplifying Expressions

The ultimate goal is to simplify an algebraic expression as completely as possible. An expression is simplified when there are no parentheses, no like terms to combine, and every term is as simplified as it can be. After distribution and combining, our expression went from:
8 + 4(3x + 6) to 32 + 12x.
Now, let's think about why this is important. Simplifying expressions makes them easier to understand and work with, especially in solving equations or further algebraic manipulations. When you simplify you:

• Reduce complexity
• Make calculations easier
• Ease further steps in solving problems (like finding variables)
• Ensure clear communication of mathematical ideas

Simplified forms are not only neat but imperative for correct and efficient calculations!