Problem 38 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(8(5 x+4)\)

Step-by-Step Solution

Answer

The simplified expression is \(40x + 32\).

Step 1: Distribute the 8

Apply the distributive property by multiplying 8 with each term inside the parentheses. This involves multiplying 8 by each term separately: 8 multiplied by 5x plus 8 multiplied by 4.

Step 2: Perform the Multiplications

Execute the multiplications from Step 1:1. Compute \(8 \times 5x\) which equals \(40x\).2. Compute \(8 \times 4\) which equals \(32\).

Step 3: Write the Expression

Combine the results from Step 2 into a single expression. Thus, the expression becomes: \(40x + 32\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like mathematical phrases that can include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Imagine variables as placeholders for numbers that are not specified, and they help in generalizing mathematical problems. In our exercise, the algebraic expression is \(5x + 4\). Here, \(5x\) is termed as a variable term because it contains the variable \(x\), and 4 is called a constant because it's just a standalone number. This expression can change in value depending on the value of \(x\).

When dealing with algebraic expressions, it's crucial to understand operations and how they apply. You can add, subtract, or multiply these terms using rules like the distributive property to manipulate and simplify expressions. Without changing the actual value, simplifications make expressions easier to manage. This translation from more complex to simpler form is an essential skill in algebra.

Next time you see an algebraic expression, remember: it's a neat way to represent math problems that can vary based on different inputs or scenarios.

Simplifying Expressions

Simplifying expressions means making them easier to understand or work with by breaking them down into simpler parts.

In our exercise, after applying the distributive property, we ended up simplifying \(8(5x + 4)\) into \(40x + 32\). What we did, essentially, was ensure each number was multiplied correctly according to the rules of multiplication for algebra.

Steps to Simplify Expressions with Distributive Property:

• Recognize when to use the distributive property.
• Apply the property: Multiply the number outside the parentheses with each term inside.
• Perform each multiplication separately before adding the results.

Once you perform these steps, you will find yourself with a more concise way to present the expression. Simplified expressions are easier to work with, especially when solving equations or further manipulating the expressions.

Prealgebra Concepts

Prealgebra is the building block for success in algebra and higher-level math. It involves gaining a grasp of key mathematical concepts that are foundational for tackling more complex algebraic problems later on.

Understanding the distributive property is one critical prealgebra concept. This property allows one to multiply a single term by others grouped in parentheses. It's written generally as: \(a(b + c) = ab + ac\).

In real-world terms, imagine you're distributing a set of 8 chocolates among two groups, where one group needs 5 chocolates per person and the other needs 4 chocolates. To ensure fairness, you multiply 8 by both needs, which is much like utilizing the distributive property.

Prealgebra concepts, like using the distributive property for simplifying expressions, offer young learners an opportunity to venture into problem-solving strategies dealing with numbers and variables. By mastering these concepts, students form the groundwork needed for high school algebra and other mathematical disciplines.