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Chapter 12: Problem 10

Evaluate each expression. $$ 6 ! \cdot 7 $$

Short Answer

Expert verified
5040

Step by step solution

01

Understand the factorial

A factorial, denoted by an exclamation point (!), means to multiply a series of descending natural numbers. For example, 6! (read as 'six factorial') means 6 脳 5 脳 4 脳 3 脳 2 脳 1.
02

Calculate 6 factorial

Evaluate 6! by multiplying the descending series of numbers: \[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \] Calculate step-by-step:\[ 6 \times 5 = 30 \]\[ 30 \times 4 = 120 \]\[ 120 \times 3 = 360 \]\[ 360 \times 2 = 720 \]\[ 720 \times 1 = 720 \]Therefore, \(6! = 720\).
03

Multiply the factorial result by 7

Now multiply the result from Step 2 by 7: \[ 720 \times 7 = 5040 \]

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Key Concepts

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

factorial notation
Factorial notation is a way to denote the product of an integer and all the integers below it. For instance, the factorial of 6 is written as 6!, which means 6 脳 5 脳 4 脳 3 脳 2 脳 1.

The exclamation mark (!) is used to indicate a factorial. In algebra, factorials are helpful in various calculations, such as combinations and permutations.

Understanding factorial notation is essential because it simplifies the representation of large numbers that involve multiple multiplications. Instead of writing out each step, you can write just one number followed by an exclamation point.
multiplying factorials
Multiplying numbers involving factorials requires understanding the calculated value of the factorial first. For example, if the question involves multiplying 6! by 7, you need to know the value of 6! before proceeding.

To multiply factorials, first determine each factorial's value individually. For instance, calculate 6! (which equals 720) before multiplying it by 7.

Once you have the factorial value, simply perform the regular multiplication: 720 (value of 6!) 脳 7, which equals 5040.

This method ensures clarity and correctness in your calculations, making complex multiplications more manageable.
step-by-step calculation
Understanding how to break down calculations into manageable steps is crucial for learning algebra.

Here is the step-by-step breakdown of the exercise:
  • First step: Find the factorial of 6
  • Second step: Multiply 6 脳 5 脳 4 脳 3 脳 2 脳 1
  • Third step: Calculate each multiplication step-by-step to get 720
After calculating 6! = 720, proceed to the next step:

Multiply 720 by 7 to get the final answer, 5040.

Breaking down problems helps simplify complex calculations and makes understanding the logic behind each step easier.

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Most popular questions from this chapter

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