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how does the left side of the equation simplify to the right one

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?

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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!

Explore the simplification process for the equation 2^n - 1 - (2^{n-1} - 1) = 2^{n-1}. Learn step-by-step how exponentiation and factoring are applied to reach the simplified form. Ideal for high school students in Grades 10-12.

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