Problem 10 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$5(3 a+2)$$

Step-by-Step Solution

Answer

The expression simplifies to \(15a + 10\) using the distributive property.

Step 1: Identifying the Problem

First, recognize that we need to apply the distributive property to the expression \(5(3a + 2)\). The distributive property states that \(a(b + c) = ab + ac\).

Step 2: Applying the Distributive Property

To apply the distributive property, multiply the outside term \(5\) by each term inside the parentheses separately. This means you will calculate \(5 \cdot 3a\) and \(5 \cdot 2\).

Step 3: Multiplying the Terms

Now perform the multiplication: \(5 \cdot 3a = 15a\) and \(5 \cdot 2 = 10\).

Step 4: Combining the Terms

Combine the terms from the multiplication to rewrite the expression: \(15a + 10\). This is your final expression after applying the distributive property.

Key Concepts

Understanding Prealgebra

Prealgebra serves as the foundation for all future algebra studies. It introduces you to the basic concepts and operations that you will use extensively in algebra. One of the key ideas is understanding expressions and how to manipulate them. Prealgebra covers arithmetic with whole numbers, fractions, decimals, and introduces simple equations and inequalities.

When working with expressions like \(5(3a + 2)\), prealgebra skills help you identify operations you can perform, such as addition and multiplication. Mastering these foundational skills is essential, as they build the groundwork for more complex algebraic concepts. It also sets the stage for using the distributive property, which is critical when dealing with expressions that contain parentheses.

Exploring Algebra Basics

Algebra basics go beyond arithmetic by introducing variables and the rules that govern their operations. A variable is a symbol, often a letter, that represents a number. In the expression \(5(3a + 2)\), \(a\) is a variable and \(5\), \(3\), and \(2\) are constants. Algebra involves working with these elements to simplify expressions and solve equations.

One of the essential skills in algebra basics is applying the distributive property. This rule is used to eliminate parentheses and simplify expressions. By understanding and applying the distributive property, you can expand expressions like \(5(3a + 2)\) to \(15a + 10\). This process makes it easier to solve more complex algebraic equations later on. It highlights how numbers and variables interact in mathematical expressions.

Mastering Multiplication in Algebra

Multiplication in algebra is crucial for simplifying expressions and solving equations. It involves multiplying numbers, variables, or both to get products. In the expression \(5(3a + 2)\), multiplication occurs between the number \(5\) and each term inside the parentheses.

To master multiplication in algebra, it’s important to understand not only how to multiply numerical values but also how to deal with terms that involve variables. For example:

• Multiply the constant outside the parentheses by each term inside: \(5 \cdot 3a = 15a\) and \(5 \cdot 2 = 10\).
• Express the result as a sum of products: \(15a + 10\).

By frequently practicing this skill, you enhance your ability to manipulate and simplify even more complex expressions, enabling you to tackle higher-level algebraic problems with ease.