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Let's simplify the given expression:

\[
\frac{e^{2x} - e^{x} + 1}{e^{x}}
\]

### Step-by-Step Solution:

1. **Break down the terms in the numerator**:
   The expression can be seen as a fraction, where we can split the terms:
   \[
   \frac{e^{2x}}{e^{x}} - \frac{e^{x}}{e^{x}} + \frac{1}{e^{x}}
   \]

2. **Simplify each term**:
   - \(\frac{e^{2x}}{e^{x}} = e^{2x - x} = e^{x}\)
   - \(\frac{e^{x}}{e^{x}} = 1\)
   - \(\frac{1}{e^{x}} = e^{-x}\)

3. **Combine the results**:
   \[
   e^{x} - 1 + e^{-x}
   \]

Thus, the simplified form of the original expression is:

\[
e^{x} - 1 + e^{-x}
\]

Would you like a more detailed explanation or have any questions?

Here are some related questions to deepen your understanding:

1. How do you simplify expressions involving exponential functions in general?
2. What are the properties of exponents that allow us to simplify \( e^{2x}/e^{x} \)?
3. Can the simplified expression \( e^{x} - 1 + e^{-x} \) be factored further?
4. What are some common mistakes to avoid when simplifying expressions like this?
5. How would you differentiate or integrate the simplified expression?

**Tip:** Remember, when dividing exponential functions with the same base, you subtract the exponents!

Simplify the expression (e^(2x) - e^x + 1) / e^x

This problem demonstrates how to simplify the expression (e^(2x) - e^x + 1) / e^x using the properties of exponents. The step-by-step solution walks through algebraic manipulation and exponential function properties. Suitable for students in Grades 10-12.

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