Problem 9 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(n+2) 3$$

Step-by-Step Solution

Answer

The equivalent expression is \(3n + 6\).

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b+c)\) is equal to \(ab + ac\). This property allows you to multiply a single term by each term inside a parenthesis.

Step 2: Apply the Distributive Property

Given the expression \((n+2)3\), identify the terms to be distributed. Here, \(3\) will be multiplied by both \(n\) and \(2\).

Step 3: Distribute 3 to Each Term

Multiply \(3\) by \(n\) to get \(3n\). Then, multiply \(3\) by \(2\) to get \(6\). Write the expression as the sum of the two products: \(3n + 6\).

Key Concepts

Algebraic Expressions

When working with algebra, you'll often encounter algebraic expressions. These expressions consist of numbers, variables, and operators such as addition and multiplication. Variables are symbols, often letters, that represent unknown or changeable values. For example, in the expression \(3n + 6\), "\(n\)" is a variable, while 3 and 6 are constants.

Algebraic expressions can be simplified or manipulated using various algebraic rules and properties, like the Distributive Property, to make them easier to work with or to solve equations. Understanding algebraic expressions is fundamental to solving mathematical problems and allows you to develop equations that model real-world scenarios.

Multiplication

Multiplication is one of the basic operations in mathematics that involves combining equal groups. It is often used to simplify repeated addition. For example, \(3 \times 2\) means adding 3 twice, which equals 6.

In algebra, multiplication can become slightly more complex when variables are involved. When a number is multiplied by a variable, like in \(3n\), it is simply an abbreviated way of expressing \(n + n + n\). This operation is crucial when applying the Distributive Property, where you need to ensure each term within parentheses is multiplied individually by the outside term.

• Helps in distributing terms over addition or subtraction inside parentheses.
• Ensures each component of the algebraic expression is accounted for.

Properly executing multiplication within algebraic expressions is critical for simplifying or solving them.

Pre-Algebra Concepts

Pre-algebra lays the foundational skills necessary to understand algebra. It involves comprehending basic mathematical concepts, such as arithmetic operations, understanding simpler expressions, and preparing for more complex algebraic tasks.

One essential concept in pre-algebra is an understanding of the properties of operations, including the Distributive Property. Knowing how to apply this property helps in simplifying expressions by efficiently managing the multiplication of terms.

• Develops skills in identifying terms and coefficients within expressions.
• Prepares students for handling equations and more advanced algebraic techniques.

Grasping pre-algebra concepts helps build confidence in tackling higher-level mathematics and develops logical thinking abilities.