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The expression we are analyzing is \(\frac{w}{w^{-4}}\). To simplify this:

\[
\frac{w}{w^{-4}} = w^{1 - (-4)} = w^{1 + 4} = w^5.
\]

Now let's examine each of the given options and compare them to \(w^5\):

- **A:** \(w^{11}\) – This is not equivalent to \(w^5\).
- **B:** \(w^{-3}\) – This is not equivalent to \(w^5\).
- **C:** \(w^3\) – This is not equivalent to \(w^5\).
- **D:** \(\left(\frac{1}{w}\right)^3 = w^{-3}\) – This is not equivalent to \(w^5\).
- **E:** \(\left(\frac{1}{w}\right)^{-3} = w^3\) – This is not equivalent to \(w^5\).
- **F:** \(\left(\frac{1}{w}\right)^{11} = w^{-11}\) – This is not equivalent to \(w^5\).

None of the provided options match the simplified form \(w^5\), so the answer should be that none of the expressions are equivalent.

Would you like to explore why certain transformations happen the way they do, or need help with another part of this? 

Here are 5 relative questions:
1. How do negative exponents work in fraction form?
2. How does the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\) apply in general?
3. What happens when you multiply or divide by a negative exponent?
4. Can exponent rules apply to other operations, like roots or logarithms?
5. How does exponentiation behave with fractions and reciprocals?

**Tip:** Always remember that subtracting a negative exponent is equivalent to adding its positive counterpart.

Which expressions are equivalent to \(\frac{w}{w^{-4}}\)? Assume w is nonzero. Select all that apply.

Learn how to simplify exponential expressions with negative exponents. This problem asks you to determine which expressions are equivalent to \( \frac{w}{w^{-4}} \), involving key exponent rules. Suitable for students in Grades 9-11.

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