In the realm of mathematics, we encounter expressions that can be evaluated in various ways, often leading to surprising or elegant solutions. Today, we delve into an intriguing expression that combines factorials and multiplication:
4! • 3! / (12 • 30 • 144 • 5,040)
This expression might appear daunting at first glance, but with a strategic approach, we can unveil its answer and explore the mathematical concepts at play.
Table of Contents
Deciphering the Expression: Factorials and Simplification
The key to evaluating this expression lies in understanding factorials and strategically simplifying the terms. Let’s break down the components:
Factorial (n!): Represented by an exclamation mark, a factorial signifies the product of all positive integers less than or equal to n. For instance, 4! (four factorial) equals 4 x 3 x 2 x 1 = 24.
Multiplication: The expression involves standard multiplication between numerical terms and factorials.
Our initial step is to tackle the factorials. We can calculate 4! and 3! separately:
- 4! = 4 x 3 x 2 x 1 = 24
- 3! = 3 x 2 x 1 = 6
Substituting these values into the expression, we get:
24 • 6 / (12 • 30 • 144 • 5,040)
Now, we can focus on simplifying the denominator. Look for common factors among the terms:
- 12 = 2 x 2 x 3
- 30 = 2 x 3 x 5
- 144 = 2 x 2 x 2 x 2 x 3 x 3
- 5,040 = 2 x 2 x 2 x 3 x 3 x 5 x 7
By carefully examining each term, we can identify factors that appear multiple times in the denominator. We can rewrite the denominator to group these common factors:
(2 x 2 x 3) • (2 x 3 x 5) • (2 x 2 x 2 x 2 x 3 x 3) • (2 x 2 x 2 x 3 x 3 x 5 x 7)
Now, we can extract these common factors outside the parentheses:
((2 x 2 x 3) • (2 x 3 x 5) • (2^4 • 3^2) • (2^3 • 3^3 • 5 • 7))
Next, we can simplify the remaining terms within each parenthesis by dividing by 1 (any number divided by 1 remains the same). We are essentially removing redundant factors of 1.
((2 • 2 • 1) • (1 • 1 • 5) • (2^4 • 1) • (1 • 1 • 5 • 7))
This leaves us with:
(4 • 5 • 16 • 35)
Unveiling the Answer and Mathematical Elegance
The final step is to multiply the remaining terms:
4 • 5 • 16 • 35 = 80 x 16 x 35 = 4,480 x 35 = 156,800
Therefore, the evaluation of the expression 4! • 3! / (12 • 30 • 144 • 5,040) yields 156,800.
This process showcases the beauty of mathematics, where seemingly complex expressions can be untangled through a methodical approach. By recognizing factorials, strategically simplifying terms, and identifying common factors, we were able to arrive at the answer.
Beyond the Answer: Exploring the Journey
The journey of evaluating this expression goes beyond obtaining a numerical answer. It highlights the significance of:
- Factorials:Understanding factorials opens doors to various areas of mathematics, including combinatorics, probability, and calculus.
- Simplification:The ability to strategically simplify expressions is a fundamental mathematical skill that proves invaluable in solving complex problems.
- Logical Reasoning:The process requires logical analysis to identify patterns and common factors, fostering critical thinking skills.
By delving into such expressions, we not only reach solutions but also sharpen our mathematical toolset and develop a deeper appreciation for the elegance and logic inherent in mathematics.
FAQs on the Expression: 4! • 3! / (12 • 30 • 144 • 5,040)
1. What does the exclamation mark (!) mean in the expression?
The exclamation mark signifies a factorial. For example, 4! (four factorial) means 4 x 3 x 2 x 1.
2. How do I solve an expression with factorials?
Calculate each factorial separately first. Then, focus on simplifying the remaining terms by finding common factors among them.
3. What’s the trick to simplifying the denominator in this expression?
Look for factors that appear multiple times in each term of the denominator. Rewrite the denominator to group these common factors and simplify by dividing them out.
4. Is there a specific order to tackle the simplification?
While the order might not be crucial here, a strategic approach is beneficial. Focus on factorials first, then identify common factors in the denominator, and finally simplify any remaining terms within parentheses.
5. Does the answer have any practical applications?
The specific answer (156,800) might not have a direct real-world application in this case. However, the process of solving the expression strengthens your understanding of factorials and simplification techniques, which are valuable in various mathematical fields.
6. What mathematical concepts are explored through this expression?
This expression touches upon factorials, multiplication, and strategic simplification. It also encourages critical thinking and logical reasoning to identify patterns and common factors.
7. What can I learn beyond solving this expression?
The journey of solving this expression goes beyond the answer. It emphasizes the importance of factorials, strategic simplification, and logical reasoning – all essential skills for mathematical problem-solving and critical thinking.
