Let's break down the simplification step by step for the equation:

$2^n - 1 - (2^{n-1} - 1) = 2^{n-1}(2 - 1) = 2^{n-1}$

Step 1: Simplify the terms inside the parentheses

The expression inside the parentheses is:

$2^{n-1} - 1$

So, the full equation becomes:

$2^n - 1 - (2^{n-1} - 1) = 2^n - 1 - 2^{n-1} + 1$

Step 2: Combine the constants

Now, combine the constants $-1$ and $+1$, which cancel each other out:

$2^n - 2^{n-1}$

Step 3: Factor out $2^{n-1}$

Notice that $2^n = 2 \cdot 2^{n-1}$, so we can factor out $2^{n-1}$ from the expression:

$2^{n-1}(2 - 1) = 2^{n-1}$

Thus, the left side simplifies to $2^{n-1}$, which matches the right side of the equation.

Would you like further clarification or more details on this simplification?

Here are five follow-up questions to expand on this:

1. What is the general rule for factoring powers of two?
2. How would the expression change if $n$ were a specific number, like 4 or 5?
3. Why do the constants $-1$ and $+1$ cancel out in this simplification?
4. What other expressions could be simplified using similar factoring techniques?
5. How does factoring out $2^{n-1}$ help in simplifying more complex equations?

Tip: When simplifying expressions with exponents, always check for common factors that can help reduce the terms easily!