https://r2.erweima.ai/mathbot/image/1726877174809q9jsr8or.jpg

The text extraction from the image was not accurate. However, from visual inspection of the image, the problem appears to be:

\[
E = \frac{2^{n+5} - 2^{n+2} - 2^{n+1}}{2^{n+4} - 2^{n+3}}
\]

Let me proceed with solving this expression step by step.

### Step 1: Simplify the numerator and denominator
First, let's factor out the powers of \(2^n\) in both the numerator and denominator.

Numerator:
\[
2^{n+5} - 2^{n+2} - 2^{n+1} = 2^n(2^5 - 2^2 - 2^1) = 2^n(32 - 4 - 2) = 2^n(26)
\]

Denominator:
\[
2^{n+4} - 2^{n+3} = 2^n(2^4 - 2^3) = 2^n(16 - 8) = 2^n(8)
\]

### Step 2: Simplify the entire fraction
Now, divide the numerator by the denominator:
\[
E = \frac{2^n(26)}{2^n(8)} = \frac{26}{8} = 3.25
\]

Thus, the value of \(E\) is:

\[
E = 3.25
\]

Would you like more details on any step or have further questions? Here are a few related questions you might explore:

1. How would the expression change if the powers of 2 were modified?
2. What is the significance of factoring out common powers of 2 in expressions like this?
3. How does this simplification method apply to more complex exponents?
4. Can you think of a case where factoring might not work easily?
5. What happens if negative powers of 2 were involved?

**Tip**: When simplifying expressions involving powers of the same base, always look to factor common terms for easier calculation!

Simplify the expression: E = (2^(n+5) - 2^(n+2) - 2^(n+1)) / (2^(n+4) - 2^(n+3)).

Discover how to simplify the exponential expression E = (2^(n+5) - 2^(n+2) - 2^(n+1)) / (2^(n+4) - 2^(n+3)) with a detailed step-by-step guide. This exercise enhances understanding of exponents and algebraic simplifications, suitable for high school students.

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