Math Problem Statement

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

Solution

Let's simplify the given expression:

e2xex+1ex\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: e2xexexex+1ex\frac{e^{2x}}{e^{x}} - \frac{e^{x}}{e^{x}} + \frac{1}{e^{x}}

  2. Simplify each term:

    • e2xex=e2xx=ex\frac{e^{2x}}{e^{x}} = e^{2x - x} = e^{x}
    • exex=1\frac{e^{x}}{e^{x}} = 1
    • 1ex=ex\frac{1}{e^{x}} = e^{-x}
  3. Combine the results: ex1+exe^{x} - 1 + e^{-x}

Thus, the simplified form of the original expression is:

ex1+exe^{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 e2x/exe^{2x}/e^{x}?
  3. Can the simplified expression ex1+exe^{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!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Algebraic Manipulation

Formulas

e^(a + b) = e^a * e^b
e^a / e^b = e^(a - b)

Theorems

Properties of Exponents

Suitable Grade Level

Grades 10-12