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Let’s analyze each expression one by one to determine if it is equivalent to \(2^9\), \(2^{-9}\), or neither.

### Expression 1: \( \frac{2^5}{2^{-4}} \)
Using the properties of exponents: \(\frac{a^m}{a^n} = a^{m - n}\).

\[
\frac{2^5}{2^{-4}} = 2^{5 - (-4)} = 2^{5 + 4} = 2^9
\]

**Answer**: \(2^9\)

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### Expression 2: \(2^3 \cdot 2^{-3}\)
Using the properties of exponents: \(a^m \cdot a^n = a^{m + n}\).

\[
2^3 \cdot 2^{-3} = 2^{3 + (-3)} = 2^0 = 1
\]

**Answer**: Neither

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### Expression 3: \((2^{-1})^9\)
Using the power of a power rule: \((a^m)^n = a^{m \cdot n}\).

\[
(2^{-1})^9 = 2^{-1 \cdot 9} = 2^{-9}
\]

**Answer**: \(2^{-9}\)

---

### Expression 4: \(\frac{1}{2^{-9}}\)
This can be rewritten using the property \(\frac{1}{a^{-m}} = a^m\).

\[
\frac{1}{2^{-9}} = 2^9
\]

**Answer**: \(2^9\)

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### Summary of Answers:
1. \( \frac{2^5}{2^{-4}} \): \(2^9\)
2. \(2^3 \cdot 2^{-3}\): Neither
3. \((2^{-1})^9\): \(2^{-9}\)
4. \(\frac{1}{2^{-9}}\): \(2^9\)

Would you like further explanation on any of these steps?

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### Related Questions:
1. How do properties of exponents simplify complex expressions?
2. What happens if you multiply expressions with the same base but different exponents?
3. How do you deal with negative exponents in a fraction?
4. What is the significance of \(2^0\) in simplifying expressions?
5. Why does the expression \(\frac{1}{a^{-m}} = a^m\) work?

### Tip:
Always apply exponent rules step-by-step to avoid mistakes, especially when dealing with negative exponents and division.

Determine whether each expression is equivalent to 2^9, 2^-9, or neither.

Learn how to determine whether expressions involving exponents are equivalent to 2^9, 2^-9, or neither. This guide covers essential exponent rules and properties for simplifying expressions.

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