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The expression provided is:

\[
F = \frac{2^{a+1} + 2^{a+2} + 2^{a+3} + 2^{a+4}}{2^{a-1} + 2^{a-2} + 2^{a-3} + 2^{a-4}}
\]

To simplify this expression, let's first factor out the common term from the numerator and the denominator.

### Numerator:
\[
2^{a+1} + 2^{a+2} + 2^{a+3} + 2^{a+4}
\]
Factor out \(2^{a+1}\):
\[
2^{a+1} (1 + 2 + 2^2 + 2^3) = 2^{a+1} (1 + 2 + 4 + 8) = 2^{a+1} \times 15
\]

### Denominator:
\[
2^{a-1} + 2^{a-2} + 2^{a-3} + 2^{a-4}
\]
Factor out \(2^{a-4}\):
\[
2^{a-4} (2^3 + 2^2 + 2 + 1) = 2^{a-4} \times 15
\]

### Now substitute back into the expression for \(F\):
\[
F = \frac{2^{a+1} \times 15}{2^{a-4} \times 15}
\]

The 15s cancel out:
\[
F = \frac{2^{a+1}}{2^{a-4}}
\]

This simplifies to:
\[
F = 2^{(a+1) - (a-4)} = 2^5 = 32
\]

So, the simplified value of \(F\) is **32**.

Would you like more details or have any questions?

### Here are 5 related questions:
1. How would the value of \( F \) change if the exponents in the numerator were increased by 1?
2. What would the expression look like if the exponents in the denominator were reduced by 2?
3. How does factoring out common terms simplify complex exponential expressions?
4. Can you explain why the powers of 2 added together led to a simple factor in this case?
5. What if the base of the exponent were different, say 3 instead of 2?

### Tip:
When simplifying expressions with exponents, always check if you can factor out common terms, as it often leads to significant simplifications.

Simplify the expression: F = (2^(a+1) + 2^(a+2) + 2^(a+3) + 2^(a+4)) / (2^(a-1) + 2^(a-2) + 2^(a-3) + 2^(a-4)).

Learn how to simplify the expression F = (2^(a+1) + 2^(a+2) + 2^(a+3) + 2^(a+4)) / (2^(a-1) + 2^(a-2) + 2^(a-3) + 2^(a-4)) by factoring common terms and applying laws of exponents. Suitable for high school students in Grades 9-12.

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