Problem 60 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (14-12)(13-8)(9-6) $$

Step-by-Step Solution

Answer

The simplified expression is 30.

Step 1: Simplify Inside Parentheses

First, simplify the expression inside each pair of parentheses: \((14-12) = 2\)\((13-8) = 5\)\((9-6) = 3\)

Step 2: Multiply the Simplified Terms Together

Next, multiply the simplified results together: \(2 \times 5 \times 3 = 30\)

Key Concepts

Numerical Expressions

Numerical expressions are mathematical statements that involve numbers and operational symbols, such as plus or minus, that can be evaluated to find their value. They can be as simple as a single number, like 7, or more complex, involving multiple operations, such as in the expression \((14-12)(13-8)(9-6)\). In this situation, the expression involves several numbers and subtraction operations, all wrapped in parentheses. The goal is to simplify the expression to find a single numerical result. To achieve this, follow specific rules and procedures that ensure each operation performed correctly results in an accurate final value.

Order of Operations

Calculating expressions correctly relies on following the proper order of operations. This set of rules helps us decide which operations to perform first, ensuring that everyone who simplifies the same expression comes to the same answer. In mathematics, this is commonly remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In problem-solving, parentheses indicate which part of an expression should be simplified first. Once inside, apply any exponents. Then, carry out multiplication and division, moving from left to right through the expression. Finally, complete addition and subtraction, also from left to right. Knowing this sequence is key to avoiding mistakes and finding the correct simplification of numerical expressions.

Parentheses Simplification

One crucial aspect of solving numerical expressions effectively is simplifying parts of the expression inside parentheses. Parentheses group together parts of the expression, indicating they should be resolved first, according to the order of operations.When you encounter expressions within parentheses, simplify them step by step. For example, with the expression \((14-12)\), subtract the two numbers inside to get 2. Repeat this method for all other parts inside parentheses, like \((13-8)\) = 5 and \((9-6)\) = 3.
This step helps reduce complex expressions into simpler forms, making them easier to handle in subsequent operations, such as multiplication or division. After simplifying the content within the parentheses, you can proceed to the next steps in the expression, which may involve other operations outside the parentheses.