Simplifying Factorials: How to Write 90 × 8! in the Simplest Form

Factorials are a common topic in mathematics, and simplifying them can often lead to elegant and efficient solutions. In this article, we will explore how to simplify the expression 90 × 8! into its simplest factorial form. This method will be straightforward and easy to follow, suitable for both beginners and experienced mathematicians.

Understanding the Problem

The expression given is 90 × 8!. Factors and prime factors play a crucial role in simplifying such expressions. Through a detailed analysis, we can break down the given expression into its simplest form.

Finding the Simplest Form of 90 × 8!

The expression 90 × 8! in its simplest factorial form is 10!. Let's go through the steps to see why this is the case.

Step-by-Step Simplification

First, we express 90 in terms of its factors:

90 9 × 10 9 3 × 3 10 2 × 5

Thus, 90 can be written as 2 × 3 × 3 × 5.

Next, we express 8 in terms of its factors:

8 2 × 2 × 2

Combining these, the prime factorization of 90 is:

90 2 × 3 × 3 × 5

The prime factorization of 8 is:

8 2 × 2 × 2

Thus, when we multiply 90 × 8!, we have:

90 × 8! (2 × 3 × 3 × 5) × (8!) 8! 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 Multiplying these, we see that the 5 and one of the 3s in 90 cancel out with the 5 and one of the 3s in 8! Therefore, the simplified form is 10 × 9 × 8! 10!

Pathetically Simple

As a result, we can simplify 90 × 8! to 10!:

90 × 8! 9 × 10 × 8! 10!

Conclusion

By understanding the prime factorization and simplifying the expression step-by-step, we can efficiently simplify 90 × 8! to 10!. This method not only makes the expression easier to understand but also helps in solving more complex problems involving factorials. Whether you're a student or a professional, mastering the art of simplifying factorials can significantly enhance your mathematical skills.