To simplify the expression \(( -5r^4s^{10} )^4\), we need to apply the power rule for exponents, which states that when you raise a power to another power, you multiply the exponents.

Let's break it down step by step:

1. **Simplify the constant**:
   \[
   (-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 625
   \]

2. **Apply the power to the variable \(r^4\)**:
   \[
   (r^4)^4 = r^{4 \times 4} = r^{16}
   \]

3. **Apply the power to the variable \(s^{10}\)**:
   \[
   (s^{10})^4 = s^{10 \times 4} = s^{40}
   \]

Now, put it all together:
\[
(-5r^4s^{10})^4 = 625r^{16}s^{40}
\]

So, the simplified expression is:
\[
625r^{16}s^{40}
\]

Do you want more details or have any questions?

Here are five related questions to expand your understanding:

1. How do you apply the power rule to negative bases with variables?
2. What happens if the exponent is odd for a negative base?
3. How does the power rule apply to multiplication of different bases?
4. How do you simplify an expression with both a negative coefficient and negative exponents?
5. Can you simplify \((2x^3y^{-5})^3\)?

**Tip**: When raising powers to powers, always multiply the exponents for each base separately.

Simplify the expression (−5r^4s^{10})^4 by applying the power rule for exponents. This step-by-step solution covers how to handle exponents on variables and negative bases. Suitable for students in Grades 8-10.

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