To simplify the expression \((2a^2b^5c^4)^5\), we will use the power rule for exponents. The power rule states that:

\[
(x^m)^n = x^{m \cdot n}
\]

Let's apply this rule to each term inside the parentheses:

1. The coefficient \(2\) raised to the power of \(5\) is:

\[
2^5 = 32
\]

2. For the variable \(a\), we have:

\[
(a^2)^5 = a^{2 \cdot 5} = a^{10}
\]

3. For the variable \(b\), we have:

\[
(b^5)^5 = b^{5 \cdot 5} = b^{25}
\]

4. For the variable \(c\), we have:

\[
(c^4)^5 = c^{4 \cdot 5} = c^{20}
\]

Now, multiplying all these together, we get:

\[
(2a^2b^5c^4)^5 = 32a^{10}b^{25}c^{20}
\]

### Final Answer:

\[
\boxed{32a^{10}b^{25}c^{20}}
\]

Would you like more details, or do you have any questions about the steps?

### Related Questions:
1. How do you apply the power rule to expressions with multiple terms?
2. What is the general form of the power rule for exponents?
3. How do you simplify expressions involving different base exponents?
4. Can you explain how to handle negative exponents in expressions like these?
5. What are some common mistakes when applying exponent rules?

### Tip:
Remember that when raising a product to a power, you must apply the exponent to each term in the product separately!

Learn how to simplify the expression (2a^2b^5c^4)^5 using the power rule for exponents. This problem covers key algebraic concepts such as applying the power rule to coefficients and variables with exponents. Suitable for students in Grades 9-10.

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