Problem 24 - Elementary Algebra

Question

For the following problems, use algebraic notataion. 8 times \((x+4)\)

Step-by-Step Solution

Answer

Question: Expand and simplify the expression 8(x+4). Answer: 8x + 32

Step 1: Identify the expression which needs to be expanded

We are given the expression: \(8(x+4)\)

Step 2: Apply the distributive property of multiplication

The distributive property states that: \(a(b+c) = ab+ac\). Using the distributive property, we distribute the 8 to both terms inside the parentheses: \(8(x+4) = 8 \cdot x + 8 \cdot 4\)

Step 3: Simplify the expression

By performing the multiplication, we simplify the expression: \(8x + 8 \cdot 4 = 8x + 32\) So, \(8(x+4) = 8x + 32\).

Key Concepts

Algebraic Notation

When diving into the world of algebra, understanding the language it speaks is crucial. Algebraic notation is this language, a system of symbols and the rules for using those symbols, that we use to express mathematical ideas and operations. It’s like learning the alphabet before you write sentences. In algebra, we use letters called variables, like 'x' and 'y', to represent numbers that might change or that we don't know yet.

For example, in the expression \(8(x+4)\),

• '8' is a coefficient, representing how many times we'll use the value within the parentheses.
• 'x' is our variable, which can stand for any number.
• '+', or plus, is an operation indicating that we're going to combine quantities.
• '4' is a constant, a specific number that won't change within the context of this problem.

By using algebraic notation correctly, we communicate and solve mathematical problems accurately and efficiently!

Multiplication in Algebra

Multiplication in algebra might seem intimidating at first, especially when it involves variables, but it's really just an extension of what you know from arithmetic. Unlike addition and subtraction, which deal with combining or removing things, multiplication involves repeated addition. It's like saying, 'I have 8 groups of something.'

In the context of our exercise, \(8(x+4)\), think of the entire parenthesis \(x+4\) as a single group. You have '8' of these groups. By using multiplication, you are essentially adding \(x+4\) together 8 times. It isn’t necessary to know what 'x' is right now; you follow the rules of multiplication to distribute the 8 across each item within the group. Multiplication in algebra follows the same rules as normal multiplication, but we also need to consider these variables.

Simplifying Expressions

The goal of simplifying expressions in algebra is to make them as 'clean' or as easy to understand as possible. This typically involves combining like terms and eliminating parentheses using properties like the distributive property, as seen in our exercise. When we simplify \(8(x+4)\), we're making the expression easier to work with.

To simplify, we first distribute the 8 to both terms inside the parentheses. We're using the distributive property, which essential says that you can 'distribute' or multiply a single term across terms inside parentheses. In our case, \(8 \times x\) and \(8 \times 4\) are calculated to get \(8x + 32\). This expression is simpler because it no longer contains parentheses, and it's written in a form that makes it clear how many x's we have and what constant is present. Simplifying is a key step in finding solutions to algebra problems.