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Understanding how factorial notation works begins with understanding integer products. An integer product is the result of multiplying two or more integers together. In the case of factorials, we are multiplying a series of integers. For example, in the expression 11!, you need to multiply all the positive integers from 11 down to 1.

Starting with 11 and multiplying downwards looks like this:
11 × 10 gives us 110
Then multiplying the result (110) by 9 gives us 990, and so on.
By continuing this series of multiplications, you will eventually get the final product.

To simplify a factorial expression, follow systematic steps to manage the large products.

First, understand what the factorial notation means. In our example, 11! means multiplying 11 by every integer less than it down to 1. Specifically, it is:
11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

Next, perform the multiplication in a step-by-step manner to simplify:
11 × 10 = 110
110 × 9 = 990
990 × 8 = 7920
And continue multiplying until you reach the product:
39916800.

Breaking down the problem into smaller steps can make it easier to manage and verify your work as you progress.

Factorial notation is an example of an algebraic expression. Algebraic expressions use letters, numbers, and symbols to represent values and quantities. Here, the '!' symbol represents the factorial operation.

Understanding how to work with algebraic expressions, like factorials, helps you simplify more complex problems. For example, if we had (n + 1)! where n = 10, this would mean:
(10 + 1)! = 11!
which reduces back to our original expression of 11!.

When simplifying factorials in algebraic expressions, it's crucial to understand the role of each component and how they interact. This knowledge will help you work more confidently with other types of algebraic expressions you may encounter.