Problem 42 - Prealgebra

Question

Simplify expression. \(3(b+2)+2 b\)

Step-by-Step Solution

Answer

The simplified expression is \(5b + 6\).

Step 1: Distribute the 3

Multiply the 3 by each term inside the parentheses, \(b+2\). This means you will perform the multiplication: \(3 \cdot b + 3 \cdot 2\).

Step 2: Simplify the Distribution

Calculate the distributed terms: \(3 \cdot b = 3b\) and \(3 \cdot 2 = 6\). This will give you the expression: \(3b + 6\).

Step 3: Combine Like Terms

Now add \(2b\) to the simplified expression \(3b + 6\). Combine the terms involving \(b\): \(3b + 2b,\) which equals \(5b\).

Step 4: Write the Final Expression

After combining like terms, the simplified expression is \(5b + 6\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps you simplify expressions, especially when dealing with parentheses. It states that when you have a number outside a set of parentheses, you can distribute, or multiply, it to each term inside the parentheses.

In the expression given: \(3(b+2) + 2b\), the first step involves using the distributive property. Here, you'll take the 3 and multiply it by each term inside the parentheses \((b + 2)\). Here's how it works:

• Multiply 3 by \(b\): \(3 \cdot b\)
• Multiply 3 by 2: \(3 \cdot 2\)

After performing these multiplications, the expression inside the parentheses is expanded into separate terms:

\(3b + 6\). This process allows you to transform a more complex grouped expression into individual parts that are easier to manage.

Combining Like Terms

After you distribute, the next step in simplifying algebraic expressions is to combine like terms.

'Like terms' are terms that have the same variable raised to the same power. In our example, we have the expression \(3b + 6 + 2b\).

To combine the like terms:

• You look for the terms that have a 'b'. These are \(3b\) and \(2b\).
• Combine them by adding their coefficients: \(3 + 2\).

This results in \(5b\). The term 6 stays as it is because it does not have a like term (it's a constant). Thus, after combining like terms, the simplified form of the expression is \(5b + 6\).

Combining like terms is crucial for simplifying expressions effectively, making calculations more straightforward and your final answers easier to interpret.

Prealgebra Concepts

Prealgebra concepts are the building blocks of algebra. They include understanding basic arithmetic and how numbers can be manipulated via operations like addition, subtraction, multiplication, and division. These concepts lay the foundation for more complex mathematical ideas.

In the problem \(3(b + 2) + 2b\), you utilize these fundamental skills:

• Multiplication through the distributive property: breaking down expressions.
• Addition for combining like terms, which is an essential skill for simplifying expressions.
• Recognizing like terms and constants helps in structuring and solving algebraic operations.

These skills are crucial in prealgebra, setting the stage for solving equations and understanding functions later on in algebra. As you become familiar with these basic techniques, your ability to handle more complicated mathematical challenges increases, making math easier and more enjoyable.