Problem 84 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ -8(3 a+2) $$

Step-by-Step Solution

Answer

Question: Simplify the expression -8(3a + 2). Answer: -24a - 16

Step 1: Understand the distributive property

The distributive property states that if a, b, and c are any numbers, then: $$ a(b + c) = ab + ac $$ We will use this property to simplify the given expression.

Step 2: Apply the distributive property

The given expression is: $$ -8(3a + 2) $$ Applying the distributive property, we get: $$ -8 \times 3a + -8 \times 2 $$

Step 3: Perform the multiplication

Now, we will perform the multiplication for each term: $$ -24a - 16 $$ The simplified expression is: $$ -24a - 16 $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and mathematical operators (like addition, subtraction, multiplication, and division). They represent mathematical relationships and can be simplified or manipulated in various ways. Let's break down the given expression

• -8(3a + 2), where 3a + 2 is an algebraic expression.
• The entire expression is part of a larger multiplication involving -8.

Understanding algebraic expressions is crucial because they are the building blocks of algebra. Each part of the expression, such as 3a and 2, has a distinct role. The term 3a represents a variable term, where 3 is a coefficient, and a is a variable. Remember, algebraic expressions can often be expanded or simplified using various algebraic rules.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. It's about making the expression as concise as possible while retaining its original value. Simplifying an expression often involves using algebraic properties, such as the distributive property, associative, and commutative laws. For the expression -8(3a + 2), simplification means performing the necessary operations to get to an equivalent expression without parentheses.

• The distributive property helps to eliminate the parentheses by distributing -8 across the terms inside.
• This leads to simplifying it to -24a - 16, a much tidier and simpler form.

This step is essential in algebra, as it not only aids in solving equations but also in understanding the relationships expressed by different terms and how they relate to one another within the expression.

Multiplication in Algebra

Multiplication in algebra is quite similar to arithmetic multiplication, but it often involves variables. This characteristic requires understanding specific rules and properties to handle it correctly. In the expression -8(3a + 2), we use multiplication in two parts:

• First, multiply -8 by 3a to get -24a.
• Second, multiply -8 by 2 to get -16.

This demonstrates how multiplication is applied element-wise in algebraic expressions, often employing the distributive property to facilitate operations on expressions within parentheses. This operation helps in transforming complex algebraic expressions into a more straightforward form, allowing easier manipulation and computation. Understanding multiplication in algebra is key, as it supports broader algebraic operations and is fundamental to solving algebraic equations.