Problem 14 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \(9+(6+x)\)

Step-by-Step Solution

Answer

The simplified expression is \(15 + x\).

Step 1: Identify the Associative Property

The associative property states that the way in which numbers are grouped in addition does not change their sum. This means that for any numbers \(a\), \(b\), and \(c\), the expression \((a + b) + c\) is equal to \(a + (b + c)\).

Step 2: Rearrange the Terms

Apply the associative property to the original expression \(9 + (6 + x)\). According to this property, you can change the grouping from \((9 + (6 + x))\) to \((9 + 6) + x\).

Step 3: Simplify the Expression

Now, perform the addition inside the parenthesis: \(9 + 6 = 15\). The expression becomes \(15 + x\).

Step 4: Write the Final Simplified Expression

The simplified form of the expression \(9 + (6 + x)\) using the associative property is \(15 + x\).

Key Concepts

Addition

Addition is one of the fundamental operations in arithmetic. It involves combining two or more numbers to get their total. In our example, we are dealing with the expression \(9 + (6 + x)\). Addition allows us to combine these values to find a sum.

• When you add numbers, order doesn't matter due to the commutative property, like in \(a + b = b + a\).
• Similarly, addition lets us use the associative property, which allows numbers to be grouped differently.

Understanding addition and its properties is crucial for simplifying expressions and performing calculations efficiently.

Grouping

Grouping in mathematics refers to the use of parentheses to indicate which operations should be performed first, especially in complex arithmetic expressions. In our task, we start with the expression \(9 + (6 + x)\).

• The numbers within parentheses, \((6 + x)\), are considered as grouped together, implying that they should be handled first when evaluating or transforming the expression.
• Through the mathematic property of association, we can adjust these groupings without altering the result.

By regrouping, as shown with \((9 + 6) + x\), we can simplify and organize calculations more effectively.

Simplification

Simplification involves reducing an expression to its most compact and easy-to-understand form. Once we apply the associative property to \(9 + (6 + x)\), we regroup it to \((9 + 6) + x\).

• The next step is to carry out basic arithmetic within the grouping: \(9 + 6 = 15\).
• After simplification, the expression becomes \(15 + x\).
• This simplified version is typically easier to work with for subsequent calculations or solving equations.

Simplifying expressions is a core skill that aids in solving mathematical problems effectively and quickly.

Prealgebra

Prealgebra is a branch of mathematics that prepares students for algebra by introducing basic mathematical concepts and operations. It covers topics like addition, subtraction, multiplication, division, and the handling of variables.

• In prealgebra, you often work with simple expressions like \(9 + (6 + x)\) to understand foundational properties such as the associative property.
• Mastering these basic concepts serves as the groundwork for more advanced subjects in mathematics.
• Practice with prealgebraic expressions helps students develop confidence and skills to tackle algebra problems in the future.

Emphasizing these fundamentals helps students build a solid mathematical foundation, ensuring their readiness for higher-level math courses.