Problem 26 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$(9+3 n) 2$$

Step-by-Step Solution

Answer

The expression $(9+3 n) 2$ when rewritten without parentheses using distributive property is \(18 + 6n\).

Step 1: Distribute the multiplier to first term

Multiply the first term inside the parentheses (which is 9) by the number outside the parentheses (which is 2), to get \(9 \times 2 = 18\).

Step 2: Distribute the multiplier to second term

Multiply the second term inside the parentheses (which is 3n) by the number outside the parentheses (which is 2), to get \(3n \times 2 = 6n\).

Step 3: Write the final expression

After distributing the number outside the parentheses to each term inside individually, place the results together to form the final expression. The resulting expression is: \(18 + 6n\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators such as addition (+), subtraction (-), multiplication (×), and division (÷). They do not have an equals sign, which differentiates them from equations. An algebraic expression can be as simple as a single number or variable, or as complex as you would like them to be. In the exercise provided, the algebraic expression is \((9 + 3n)\). Here, the expression contains a constant (9) and a term that combines a number and a variable (3n).

Variables typically represent unknown values and are denoted by letters, like \(n\) in this expression. When working with algebraic expressions, the goal is often to simplify or manipulate them using properties and operations until they are in a more useful or understandable form.

Understanding the components of algebraic expressions is fundamental for simplifying them through operations like the distributive property, which we are focusing on in this problem.

Simplifying Expressions

Simplifying expressions involves rewriting an algebraic expression in a more concise form. This often means reducing the expression into its simplest terms by removing any parentheses and combining like terms. The distributive property is a crucial tool in simplifying expressions because it allows you to multiply a single term by each term within the parentheses.

In the given exercise, you are tasked with simplifying the expression \((9 + 3n)2\) using the distributive property. This means you'll distribute the 2 across each term in the parentheses:

• Multiply the constant term 9 by 2 to get 18
• Multiply the term \(3n\) by 2 to get \(6n\)

After these independent calculations, you place 18 and \(6n\) together to form the simplified expression \(18 + 6n\). By eliminating the parentheses and combining the results, your expression becomes more straightforward and easier to work with in future algebraic operations.

Multiplication in Algebra

Multiplication in algebra follows similar rules as arithmetic multiplication, but it can involve a mix of numbers and variables. It often requires the use of the distributive property, especially when dealing with expressions that contain parentheses.

The distributive property asserts that multiplying a number by an expression in parentheses involves multiplying that number by each term inside the parentheses separately.

For the expression \((9 + 3n)2\), you'll carry out multiplication as follows:

• First, multiply 9 (a constant) by 2, resulting in 18
• Then, multiply \(3n\) by 2, yielding \(6n\)

Understanding these steps ensures clarity when simplifying or expanding expressions. This foundational skill in algebra is pivotal, allowing you to manipulate and solve more complex algebraic problems effectively.