Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$(5-2) !$$

Step-by-Step Solution

Answer

The value of \((5-2)!\) is 6.

Step 1: Simplify Inside the Parentheses

First, evaluate the expression inside the parentheses: \(5 - 2\). This simplifies to \(3\).

Step 2: Calculate the Factorial

Next, calculate \(3!\), which is the factorial of 3. The definition of factorial for a number \(n\) is the product of all positive integers less than or equal to \(n\). Thus, we calculate: \[ 3! = 3 \times 2 \times 1 = 6 \]

Step 3: Final Answer

After calculating the factorial, we find that the answer to the expression \((5-2)!\) is \(6\).

Key Concepts

Order of Operations

When tackling mathematical expressions, it’s crucial to follow the order of operations, ensuring each calculation is performed in the right sequence. This commonly follows the PEMDAS/BODMAS rule, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

In our original exercise, we started by simplifying what was inside the parentheses first, leading us to solve **\(5 - 2\)** initially. The rule dictates that expressions inside parentheses are simplified before moving on to other operations. This step-by-step approach prevents errors that could arise from performing calculations out of order.

Remember, a small error in following the order can lead to completely different results. So, practicing the order of operations ensures you're always on the right path to solving complex expressions.

Simplifying Expressions

Simplifying expressions is about breaking down the math problem into its simplest form. This helps to make the calculation process easier and more manageable. By simplifying the expression \((5-2)!\), we handle the expression inside the parentheses first, resulting in a simpler number, which was 3.

This process involves reducing complex expressions by tackling parentheses, like in the exercise given. We simplified it to a single integer, which then allowed us to proceed with finding the factorial in a less complicated form. Each step of simplifying plays a crucial role in ensuring the operation is clear and straightforward, making further calculations easier to manage.

Practicing different types of expressions and their simplifications—it may involve combinations of terms, different operations, and operations of degrees—helps improve one's math skills and confidence.

Precalculus

Precalculus involves understanding fundamental concepts, paving the way for more advanced studies in calculus. It includes a variety of topics, such as sequences, series, and functions, but also basic principles like factorials.

In the exercise given, calculating a factorial is an essential mathematical operation often seen in probability calculations or permutations. Factorials are an integral part of combinatorics and sequences,

• To calculate a factorial, multiply a sequence of descending natural numbers down to 1.
• Thus, the factorial of 3, expressed as \(3!\), is computed by multiplying 3 by all preceding positive integers: \(3 \times 2 \times 1 = 6\).

Understanding these fundamental operations is vital for success in precalculus and lays a solid foundation for upcoming calculus challenges. By mastering such basics, students prepare themselves for more intricate mathematical analyses in the future.