Problem 39 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(6(3 a+1)\)

Step-by-Step Solution

Answer

The simplified expression is \(18a + 6\).

Step 1: Identify the Expression

The given expression to simplify is \(6(3a + 1)\). This expression consists of a number (6), multiplied by a sum inside the parenthesis \((3a + 1)\).

Step 2: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). In this case, distribute the 6 to both terms inside the parenthesis: \(6 \times 3a + 6 \times 1\).

Step 3: Perform the Multiplications

Calculate each multiplication separately: \(6 \times 3a = 18a\) and \(6 \times 1 = 6\). Write these results as a new expression: \(18a + 6\).

Step 4: Simplify the Expression

Since there are no like terms to combine in \(18a + 6\), this expression is already in its simplest form.

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions can make complex problems much more manageable. When we simplify an expression, we rewrite it in a form that is easier to understand and work with, without changing its value.
One common method used is applying the distributive property. Let's look at how it was used in the expression from the exercise:

• Start with the expression: \(6(3a + 1)\).
  Here, the goal is to distribute the 6 across each term inside the parenthesis.
• Perform the multiplication: \(6 \times 3a\) results in \(18a\) and \(6 \times 1\) gives \(6\).
  This step requires applying the basic multiplication rules.
• Combine the results to get the new expression: \(18a + 6\).

This process of distribution, multiplication, and combination simplifies the original expression to make it more straightforward for future calculations.

Prealgebra

Prealgebra acts as a bridge between basic arithmetic and more advanced algebraic concepts. It introduces foundational elements such as variables, expressions, and basic properties like the distributive property. Let's understand the key aspects involved:

• Variables: These are symbols, often letters, used to represent numbers. In our exercise, 'a' is a variable.
• Expressions: These contain numbers, variables, and operators, like addition or multiplication. The expression \(6(3a + 1)\) is a great example.
• Distributive Property: A crucial rule in algebra that helps in breaking down expressions for easier handling.

By grasping these core elements in prealgebra, students build a strong foundation for tackling more complex algebraic tasks later on.

Applying Mathematical Properties

Mathematical properties like the distributive property are essential tools in algebra. They help transform and simplify expressions, which can lead to easier calculations and solutions. In the case of the expression \(6(3a + 1)\), we applied the distributive property:

• Distributive Property: This property states \(a(b + c) = ab + ac\). By using this, we expand the expression to \(18a + 6\).
• Application in Problem Solving: Such properties allow us to break down problems into manageable steps, which is invaluable in solving equations or inequalities.
• Generalization: Recognizing these properties helps in understanding the structure and logic behind algebraic operations, paving the way to solve complex math problems efficiently.

Practicing these properties regularly will improve problem-solving skills and deepen mathematical comprehension.