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The expression provided is:

\[
\sqrt{81 y^{21}}
\]

We are asked to simplify this expression, assuming \( y \) is a positive real number.

### Step-by-Step Simplification:

1. **Rewrite the square root**: 
   \[
   \sqrt{81 y^{21}} = \sqrt{81} \cdot \sqrt{y^{21}}
   \]

2. **Simplify the square root of 81**:
   \[
   \sqrt{81} = 9
   \]

3. **Simplify the square root of \( y^{21} \)**: 
   Using the property \(\sqrt{a^n} = a^{n/2}\), we can simplify:
   \[
   \sqrt{y^{21}} = y^{21/2} = y^{10.5}
   \]

4. **Final Simplified Expression**:
   \[
   9y^{10.5}
   \]

Thus, the simplified form is:

\[
9y^{10.5}
\]

Would you like further details on any step, or do you have any additional questions?

### Related Questions:
1. How do you simplify expressions involving higher roots?
2. What happens if \( y \) is not a positive real number in this case?
3. Can you express \( y^{10.5} \) using radicals instead of exponents?
4. How do you handle fractional exponents in algebraic expressions?
5. What if the base \( 81 \) were not a perfect square, how would you simplify the expression?

### Tip: 
When simplifying square roots, it's useful to remember that fractional exponents can often make the process more straightforward.

Simplify the expression \(\sqrt{81 y^{21}}\), assuming y represents a positive real number.

Simplify the radical expression \( \sqrt{81 y^{21}} \) step by step, using properties of exponents and square roots. This detailed guide breaks down the process into manageable steps, making it suitable for students in Grades 9-11.

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