Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the factorial. $$ \left(\frac{12}{6}\right) ! $$

Step-by-Step Solution

Answer

The factorial is 2.

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{12}{6} \). The division \( 12 \div 6 \) simplifies to 2. So, \( \frac{12}{6} = 2 \).

Step 2: Calculate the Factorial

Now that we have simplified the fraction to \( 2 \), calculate the factorial \( 2! \). By definition, \( 2! = 2 \times 1 = 2 \).

Key Concepts

Simplification

Simplification is a fundamental concept in mathematics that involves making expressions easier to work with without changing their value. In the context of this exercise, simplification involves reducing a fraction to its lowest terms. For example, the fraction \( \frac{12}{6} \) can be simplified by dividing both the numerator (12) and the denominator (6) by their greatest common divisor, which is 6. This gives us:

• \( 12 \div 6 = 2 \)
• \( 6 \div 6 = 1 \)

So, \( \frac{12}{6} = 2 \). This step is crucial because it transforms the expression into a simpler form, allowing us to easily compute a subsequent operation, such as finding a factorial.

Division

Division is one of the basic arithmetic operations that involves splitting a number into equal parts. It is represented by the '÷' symbol or a fraction bar. In terms of fractions, division is used to simplify expressions by determining how many times the denominator can fully divide into the numerator.
Using our example, \( 12 \div 6 \) asks, "How many times does 6 go into 12?" The answer is 2, because 6 times 2 equals 12. Understanding division in this context helps identify relationships between numbers and simplify mathematical expressions.
When dealing with fractions, always check if the numerator and denominator have a common factor, which allows you to simplify the fraction by division. This process leads directly to the next important concept in evaluating numerical expressions.

Numerical Expressions

Numerical expressions consist of numbers that define a particular value or statement, often involving operations like addition, subtraction, multiplication, and division. Reducing fractions and calculating factorials fall under the key operations performed on numerical expressions.
In this exercise, once \( \frac{12}{6} \) is simplified to 2, we proceed with evaluating the factorial of this numerical expression: \( 2! \). The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \). In our example, the factorial of 2, denoted by \( 2! \), is calculated as:

• \( 2 \times 1 = 2 \)

Thus, the numerical expression \( (\frac{12}{6})! \) simplifies and evaluates to 2 through the combination of division and factorial operation. Understanding how to handle numerical expressions is essential in problem-solving and evaluating different mathematical scenarios.