Problem 60 - Algebra for College Students

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 the First Parenthesis

Start by simplifying the expression inside the first set of parentheses: \(14 - 12\). Subtract 12 from 14 to get: \(2\).

Step 2: Simplify the Second Parenthesis

Next, simplify the expression inside the second set of parentheses: \(13 - 8\). Subtract 8 from 13 to get: \(5\).

Step 3: Simplify the Third Parenthesis

Now, simplify the expression inside the third set of parentheses: \(9 - 6\). Subtract 6 from 9 to get: \(3\).

Step 4: Multiply the Results

Multiply the results from the simplified parentheses: \(2 \times 5 \times 3\). First, multiply 2 and 5 to get 10. Then, multiply 10 by 3 to get 30.

Key Concepts

Order of Operations

When simplifying a numerical expression, it's crucial to follow a consistent set of rules known as the order of operations. These rules help determine which mathematical operations to perform first, ensuring that everyone solves the expression the same way and gets the same result. The order of operations is often remembered by the acronym PEMDAS, which stands for:

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

Parents come first! When you look at an expression, always start by simplifying inside any parentheses. This helps to organize complex calculations.
Next, you handle exponents. Following this, multiplication or division should be done as they appear from left to right. Finally, you tackle addition or subtraction. Understanding and applying these steps is key to correctly simplifying and solving numerical expressions.

Parentheses in Mathematics

Parentheses play a pivotal role in mathematics by grouping numbers and operations together, indicating that whatever is inside should be calculated first. This means, whenever you see a group of numbers inside parentheses, simplify them before moving on with the other parts of the expression.
In our exercise, expressions like \((14-12)\), \((13-8)\), and \((9-6)\) are enclosed in parentheses to be simplified at the start. By solving each of these separately first, you prevent errors and ensure the operations are done correctly.
Once the expressions inside the parentheses are simplified, the parentheses can be removed, and you can flow into the subsequent operations with ease.

Multiplication of Numbers

Once you've simplified all expressions within parentheses, you're ready to multiply the results. Multiplication is the process of scaling one number by another. In our original exercise, after simplifying each parenthesis, we end up with 3 numbers: 2, 5, and 3.
Here's how you handle multiplication step-by-step:

• First, multiply the first two numbers. So, you multiply 2 by 5, which gives you 10.
• Next, take the result (10) and multiply it by the remaining number (3). This results in 30.

Employing multiplication effectively is crucial for solving expressions and equations. It helps in quickly aggregating results from smaller calculations into a final simplified answer. Always remember, it’s important to follow the sequence to ensure accuracy.