Problem 28 - Prealgebra

Question

Apply the associative property to expression, and then simplify the result. \((14 x+3)+15\)

Step-by-Step Solution

Answer

The simplified expression is \(14x + 18\).

Step 1: Identify the Parts of the Expression

Observe the given expression: \((14x + 3) + 15\). This expression consists of two groups within the parentheses and the number 15 outside the parentheses.

Step 2: Apply the Associative Property

The associative property of addition states that the way numbers are grouped in an expression does not change the sum. Therefore, you can regroup the numbers as follows: \(14x + (3 + 15)\).

Step 3: Simplify Inside the Parentheses

Now, focus on the expression inside the parentheses: \(3 + 15\). Add these numbers together to obtain \(18\). Thus, the expression becomes \(14x + 18\).

Key Concepts

Expression Simplification

Expression simplification is like cleaning up a messy room. You're organizing and making it easier to understand what you have. In mathematics, to simplify an expression means to reduce it to its simplest form.
By simplifying, the expression becomes more straightforward to work with. Simplification can help you solve equations more easily and reduces the chance of making mistakes.

• For example, when you see an expression like \(14x + (3 + 15)\), you can perform operations inside parentheses and combine like terms.
• This process cleverly reduces the expression to \(14x + 18\).

Simplifying expressions often involves combining like terms, removing unnecessary parentheses, and sometimes using rules like the distributive or associative property.
However, always remember to perform operations according to the mathematical order of operations (PEMDAS/BODMAS) to ensure you're simplifying correctly.

Prealgebra

Prealgebra is like the on-ramp to more advanced math. It prepares you for algebra by introducing fundamental math concepts. In prealgebra, you get familiar with basic arithmetic operations and how they apply to algebraic expressions.
You'll encounter terms like 'variable', 'constant', and 'coefficient', which are essential as you start to solve equations and explore mathematical relationships.

• The expression \(14x + 18\) includes a variable 'x' with a coefficient 14, showing how prealgebra lays the foundation for understanding expressions and equations.
• Here, simplification helps reinforce skills needed for solving equations you'll see later in algebra.

Prealgebra focuses on developing a solid grounding in these skills, helping you transition smoothly to algebra, where you'll work with more complex equations and concepts.

Parentheses in Math

Parentheses in math play a crucial role in guiding how you solve expressions. They indicate which part of the expression should be calculated first. This helps you follow the correct order of operations.
When you see parentheses, it implies a priority in calculation, calling attention to specific calculations that need to be done before others.

• In \( (14x + 3) + 15 \), the parentheses tell you to focus first on calculating \(14x + 3\) together.
• Sometimes, you might need to apply properties like the associative property to regroup the terms for easier calculation, like turning the expression into \( 14x + (3 + 15) \).

The associative property lets you shift the grouping of numbers without changing their sum. This property is valuable when simplifying expressions because it offers flexibility in calculation.
Always remember: When simplifying expressions, parentheses are your roadmap to ensuring calculations are done accurately according to priority.