Problem 40 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(4(8 a+3)\)

Step-by-Step Solution

Answer

The simplified expression is \(32a + 12\).

Step 1: Apply the Distributive Property

To apply the distributive property, multiply the term outside the parentheses by each term inside the parentheses. This means we multiply 4 by each term inside, which are \(8a\) and \(3\). So, we have:\[ 4 imes 8a + 4 imes 3 \]

Step 2: Perform the Multiplications

Now, we perform the multiplication for each term:- Multiply 4 by \(8a\): \(4 imes 8a = 32a\)- Multiply 4 by 3: \(4 imes 3 = 12\)

Step 3: Combine the Results

Combine the terms to write the expression:\[ 32a + 12 \] This is the simplified form of the expression after applying the distributive property.

Key Concepts

Simplifying Expressions

When simplifying expressions in algebra, our goal is to reduce the expression to its simplest form while maintaining its meaning. This is an essential skill in algebra because it makes equations easier to work with and understand.
To simplify an expression like \(4(8a + 3)\), we utilize the distributive property—a key algebraic rule. After applying the distributive property and doing the necessary calculations, the expression is broken down into simpler components, which we then combine into a concise form.

• Apply principles like the distributive property to "open up" expressions.
• Perform any possible arithmetic operations like addition, subtraction, multiplication, or division.
• Combine like terms, which are terms that have the same variables raised to the same powers.

The result of these steps is a more straightforward expression \(32a + 12\), which is easier to use in further calculations or to understand in the context of a larger problem.

Multiplication

In algebra, multiplication isn't just about repeating addition; it's a fundamental operation we use to transform and simplify expressions.
When we see an expression such as \(4(8a + 3)\), multiplication rules guide us in correctly distributing and calculating the terms involved.
Here's the process:

• First, identify the multiplier outside the parentheses. In this example, it's \(4\).
• Then, apply this multiplier to each term within the parentheses. For \(8a\), we calculate \(4 \times 8a = 32a\).
• Similarly, for the constant \(3\), we compute \(4 \times 3 = 12\).

These calculations demonstrate how multiplication can reduce expressions and help us systematically simplify them by cleaning up complex terms into simpler parts.
Ultimately, through multiplication, we can manipulate and mold algebraic expressions, turning them into forms that are much easier to work with.

Algebra Basics

Algebra is the language of mathematics where letters and symbols represent numbers. Understanding algebra basics is crucial as they provide the foundation for more complex mathematical concepts.
One fundamental concept in algebra is using variables, like \(a\), to represent unknown values. In the expression \(4(8a + 3)\), \(a\) is our variable, and the expression translates into numbers through operations.
Key components involved here are:

• Variables: Symbols (like \(a\)) that stand in for unknown values.
• Constants: Fixed values (like \(3\)) in expressions.
• Operations: Processes like addition, subtraction, multiplication, and division that help us manipulate expressions.
• The Distributive Property: A crucial rule that helps us deal with expressions within parentheses.

Understanding these basics allows us to interpret and solve expressions and equations. It empowers us to translate real-world problems into mathematical language, making them easier to analyze and solve through logical steps.