To simplify \((5y^4)^{-2}\), we can apply the properties of exponents. The expression can be rewritten as:

\[
(5y^4)^{-2} = \frac{1}{(5y^4)^2}
\]

Now, let's square the term in the denominator:

\[
(5y^4)^2 = 5^2 \cdot (y^4)^2 = 25y^8
\]

Putting it all together, we have:

\[
(5y^4)^{-2} = \frac{1}{25y^8}
\]

Thus, the simplified form is:

\[
\frac{1}{25y^8}
\]

Would you like further details or explanations on this topic? Here are some related questions you might consider:

1. What are the general rules for simplifying expressions with negative exponents?
2. How would you evaluate \((5y^4)^{-2}\) if \(y = 2\)?
3. Can you explain how to handle exponents with products?
4. What happens if you have a negative exponent applied to a fraction?
5. How do you apply the power of a power rule to exponents?

**Tip:** Always remember that a negative exponent indicates a reciprocal.

Learn how to simplify the expression (5y^4)^-2 using rules for negative exponents and power of a power. This step-by-step solution explains each part of the process and is ideal for students in Grades 9-10.

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