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Chapter 14: Problem 10

Simplify. $$8 !$$

Short Answer

Expert verified
8! = 40320

Step by step solution

01

Understand Factorials

A factorial, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to a given positive integer. For example, the factorial of 8 (written as 8!) means multiplying 8 by every positive integer less than 8.
02

Write the Factorial Expression

Write down the expression for 8!. This is: 8! = 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
03

Perform the Multiplication Step-by-Step

Start multiplying the numbers step-by-step to avoid mistakes: 1. 8 脳 7 = 56 2. 56 脳 6 = 336 3. 336 脳 5 = 1680 4. 1680 脳 4 = 6720 5. 6720 脳 3 = 20160 6. 20160 脳 2 = 40320 7. 40320 脳 1 = 40320
04

Write the Final Answer

After performing the multiplication, we find that 8! = 40320.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factorial definition
A factorial is a mathematical concept that involves multiplying a series of descending natural numbers. It's represented by an exclamation mark (!). When you see a number followed by '!', this tells you to multiply that number by every whole number less than it, down to 1. For example, the factorial of 8 is written as 8! and means you need to multiply 8 by each integer below it (8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1). Factorials are widely used in math, especially in permutations, combinations, and in analysis.
multiplication
Multiplication is one of the four elementary operations in arithmetic. In essence, it's repeated addition. When dealing with factorials, you'll often perform a series of multiplications in sequential order. For example, to calculate 8!, start with 8 and multiply it by 7, then take that result and multiply by 6, and so on until you reach 1. Multiplication can seem daunting with larger numbers, but breaking it down step-by-step, as we do with factorials, makes it manageable.
step-by-step calculation
Breaking down calculations into smaller steps helps in understanding and ensures accuracy. Let's simplify 8! step-by-step. First, understand that 8! means '8 factorial,' or 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
  • Start with 8 脳 7 = 56
  • Next, multiply 56 by 6 to get 336
  • Continue by multiplying 336 by 5, resulting in 1680
  • Then, 1680 脳 4 = 6720
  • After that, 6720 脳 3 gives you 20160
  • Multiply 20160 by 2 to get 40320
  • Finally, multiplying 40320 by 1 leaves you with 40320
By following these steps, we find that 8! equals 40320. Step-by-step calculations help avoid mistakes and ensure that each multiplication is performed correctly before moving on to the next.

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