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Let's carefully analyze the steps shown in the expression. The expression is:

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

We want to see how \( e^{2x} \) converts into \( e^{4x} \).

1. In the third step, we multiply both the numerator and denominator of the fraction by \( e^{2x} \) to eliminate the terms with \( e^{-2x} \). This is a common algebraic technique to simplify the expression.

### Step-by-Step Simplification:

#### Starting expression:
\[
\frac{e^{2x} - \frac{1}{e^{2x}}}{e^{2x} + \frac{1}{e^{2x}}}
\]

#### Multiply both numerator and denominator by \( e^{2x} \):

- **Numerator**:
\[
e^{2x} \left( e^{2x} - \frac{1}{e^{2x}} \right) = e^{4x} - 1
\]

- **Denominator**:
\[
e^{2x} \left( e^{2x} + \frac{1}{e^{2x}} \right) = e^{4x} + 1
\]

#### Resulting expression:
\[
y = \frac{e^{4x} - 1}{e^{4x} + 1}
\]

Thus, the term \( e^{2x} \) transforms into \( e^{4x} \) by multiplying both the numerator and the denominator by \( e^{2x} \).

Would you like more details on this simplification, or do you have any other questions?

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Here are 5 related questions to deepen understanding:

1. How does multiplying both numerator and denominator by the same term affect a fraction?
2. What are the rules of exponents used in this simplification?
3. How would the expression change if we multiplied by a different power of \( e \)?
4. Can we simplify this further by introducing hyperbolic functions like \( \tanh(x) \)?
5. What happens to this function as \( x \to \infty \)?

**Tip:** Multiplying both the numerator and denominator by a common factor is a useful technique when trying to eliminate negative exponents or simplify complex fractions.

Learn how the term e^2x can be simplified to e^4x in a complex fraction. This step-by-step breakdown covers algebraic techniques and exponent rules, providing a detailed solution for high school students. Suitable for Grades 10-12.

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