Problem 37 - Prealgebra

Question

Simplify: \(15+(b+3)\)

Step-by-Step Solution

Answer

The simplified expression is \(18 + b\).

Step 1: Identify and Rewrite the Expression

The given expression is \(15 + (b + 3)\). Our goal is to simplify this expression. Begin by identifying the terms inside and outside the parentheses.

Step 2: Apply the Distributive Property

Since the expression within the parentheses is \(b + 3\), and adding it does not entail distribution as the parentheses suggest plain addition, remove the parentheses directly: \(15 + b + 3\).

Step 3: Combine Like Terms

Now, combine the like terms. The numbers 15 and 3 are like terms. Add them together to simplify the expression: \((15 + 3) + b\), which simplifies to \(18 + b\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps students to remove parentheses and simplify expressions. The property states that multiplying a number by a sum is the same as doing each multiplication separately. In formula terms, it is written as:

• \(a(b + c) = ab + ac\)

This property is used when the expression inside the parentheses needs to be distributed across each term separately. However, in the case of our given problem, \(15+(b+3)\), there is no multiplication occurring outside the parentheses, so the distributive property does not directly apply.
This exercise cleverly introduces the concept but ultimately requires a simple step of removing parentheses, preparing students for future, more complex problems where distribution is necessary.

Combining Like Terms Simplified

Combining like terms is the process of consolidating terms that have the same variable or are constants. This helps in simplifying expressions further. In the expression \(15 + b + 3\), the numbers 15 and 3 are like terms. Like terms are terms that contain the same variable, raised to the same power.

Here's how you combine them:

• Add directly the numbers together since they both do not have any variable associated with them.
• Maintain the variable term as it is, since there is no other term with \(b\) to combine with.
• The final result is the simplified expression: \(18 + b\).

This process not only helps in making equations easier to handle but also strengthens understanding of how different parts of an expression relate to each other.

The Role of Parentheses in Algebra

Parentheses are used in algebra to group terms or indicate the order of operations. They clarify which operations should be performed first and help in organizing expressions to avoid errors. When solving or simplifying algebraic expressions, different rules apply depending on the operation:

• If parentheses are preceded by a multiplication or division, use the distributive property to eliminate them.
• If they are part of a simple addition or subtraction, as in \(15 + (b + 3)\), you can directly remove them.

Understanding the role of parentheses helps prevent mistakes related to the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This clarity aids in tidying expressions and solving problems efficiently.