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The given expression is:

\[
\sqrt{16r^4s^{12}}
\]

We need to simplify this expression for all values of \(r\) and \(s\).

### Step-by-step simplification:

1. **Simplify the square root**:
   \[
   \sqrt{16r^4s^{12}} = \sqrt{16} \cdot \sqrt{r^4} \cdot \sqrt{s^{12}}
   \]
   
2. **Evaluate each part**:
   - \(\sqrt{16} = 4\)
   - \(\sqrt{r^4} = r^2\) (because \(\sqrt{r^4} = (r^4)^{1/2} = r^{4/2} = r^2\))
   - \(\sqrt{s^{12}} = s^6\) (because \(\sqrt{s^{12}} = (s^{12})^{1/2} = s^{12/2} = s^6\))

3. **Combine the simplified parts**:
   \[
   \sqrt{16r^4s^{12}} = 4r^2s^6
   \]

Therefore, the expression simplifies to \( 4r^2s^6 \).

### Answer:
The correct answer is:

\[
\boxed{4r^2s^6}
\]

Let me know if you'd like more details on any part of this solution!

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Here are some related questions to think about:

1. How would you simplify \(\sqrt{25r^6s^8}\)?
2. What happens if the exponent on \(r\) is odd, e.g., \(\sqrt{r^5}\)?
3. Can you simplify \(\sqrt{49r^8s^4}\)?
4. What is the cube root of \(8r^6s^{9}\)?
5. How would the simplification change if \(r\) and \(s\) were negative?

**Tip**: When simplifying square roots, remember that even exponents inside the square root result in whole powers outside the root.

Which expression is equivalent to √(16r^4s^12), for all values of r and s where the expression is defined?

Learn how to simplify the algebraic expression √(16r^4s^12) with a detailed step-by-step solution. This problem covers key concepts in exponents, radicals, and simplification. Suitable for students in Grades 8-10.

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