https://file.aiquickdraw.com/mathbot/file/1727709500300kcq3ms4d.jpg

The given equation is:

\[
6^{1} + 1 \times x! = 36
\]

Let's break this down step by step:

1. \(6^1 = 6\) (since any number to the power of 1 is itself).
   
   So, the equation becomes:
   \[
   6 + 1 \times x! = 36
   \]

2. \(1 \times x! = x!\) (multiplying by 1 does not change the value).

   Therefore, the equation simplifies to:
   \[
   6 + x! = 36
   \]

3. Subtract 6 from both sides to isolate \(x!\):
   \[
   x! = 36 - 6
   \]
   \[
   x! = 30
   \]

Now, we need to find the value of \(x\) such that \(x!\) (factorial of \(x\)) equals 30. By checking factorial values:

- \(4! = 4 \times 3 \times 2 \times 1 = 24\)
- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

There is no integer \(x\) such that \(x! = 30\) because the factorials jump from 24 (\(4!\)) to 120 (\(5!\)). Hence, the equation has no solution for an integer \(x\).

Would you like further details on the process, or do you have any questions?

### Related Questions:
1. What is the factorial function, and how does it grow?
2. Can factorials be solved for non-integer values?
3. How do factorials relate to permutations and combinations?
4. What are the properties of exponential and factorial functions?
5. What is Stirling’s approximation for large factorials?

**Tip:** Factorials grow very rapidly, so for larger numbers, approximation methods like Stirling's approximation are useful.

Розв'язати рівняння: 6^1 + 1 × x! = 36

Explore the step-by-step solution of the equation 6^1 + 1 × x! = 36. Understand exponential functions and factorials with a detailed explanation, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation