Problem 40 - College Algebra

Question

Evaluate the factorial. $$\left(\frac{12}{6}\right) !$$

Step-by-Step Solution

Answer

The value is 2.

Step 1: Simplify the Fraction

The first step is to simplify the fraction \( \frac{12}{6} \). When you divide 12 by 6, the result is 2. Hence, \( \frac{12}{6} = 2 \).

Step 2: Calculate the Factorial

Now that we have simplified the fraction to 2, we need to calculate \( 2! \), which is the factorial of 2. The factorial of a number \( n \), written as \( n! \), is the product of all positive integers up to that number \( n \).

Step 3: Perform the Multiplication

To calculate \( 2! \), we multiply the positive integers up to 2. So, \( 2! = 2 \times 1 = 2 \). Thus, the value of the factorial \( (\frac{12}{6})! \) is 2.

Key Concepts

Simplifying Fractions

To simplify a fraction, you need to divide both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). The GCD is the largest number that can evenly divide both numbers without leaving a remainder.

In the given exercise, the fraction is \( \frac{12}{6} \). You can find the GCD of 12 and 6, which is 6, by inspecting the two numbers. Once you know the GCD, divide both 12 and 6 by 6.

• Numerator: \( 12 \div 6 = 2 \)
• Denominator: \( 6 \div 6 = 1 \)

Now, the fraction simplifies to \( \frac{2}{1} \), which is simply 2. Simplifying fractions is crucial in making calculations easier and more efficient. In any problem, simplifying first can often simplify your work later on.

Factorial Calculation

A factorial is a mathematical operation that computes the product of all positive integers up to and including a given number. The notation \( n! \) denotes the factorial of the number \( n \). For example, the factorial of 3, written as \( 3! \), is calculated as \( 3 \times 2 \times 1 = 6 \).

Factorial calculations grow rapidly. Even small numbers can lead to large results. For this reason, it's vital to simplify any expressions before computing factorials, wherever possible.Here, after simplifying \( \frac{12}{6} \) to 2, you are asked to find \( 2! \). The calculation is straightforward:

• Start with 2: \( 2 \times 1 = 2 \)

Thus, \( 2! \) is equal to 2. Understanding how to calculate factorials is useful, particularly in areas such as statistics and probability, where these operations frequently appear.

Basic Algebra Concepts

Algebra involves using letters and symbols to represent numbers and quantities in formulas and equations. It's a step beyond basic arithmetic with the introduction of these variables.

In the context of this exercise, you are applying basic algebra principles by simplifying a fraction and calculating a factorial. Both operations are foundational skills in algebra that help solve more complex mathematical problems in the future. Here's how these concepts connect:

• Simplifying Fractions: Reduces the complexity of expressions.
• Factorial Calculation: Requires basic multiplication, an essential skill in algebra.

By understanding these individual concepts, you can tackle a wide range of algebraic problems. This foundational knowledge is essential not only for high school math but also in disciplines like computer science and engineering. Practicing these core concepts will make you more comfortable with algebra and ready to face more advanced topics.