https://file.aiquickdraw.com/mathbot/file/1727011814042foypqr1n.jpg

The given expression to simplify is:

\[
\left( 81 y^{16} \right)^{\frac{3}{4}}
\]

### Step-by-step simplification:

1. **Apply the exponent rule:**  
   When raising a product to a power, you can distribute the exponent to both terms:
   \[
   (a \cdot b)^n = a^n \cdot b^n
   \]
   So:
   \[
   \left( 81 y^{16} \right)^{\frac{3}{4}} = 81^{\frac{3}{4}} \cdot \left( y^{16} \right)^{\frac{3}{4}}
   \]

2. **Simplify \(81^{\frac{3}{4}}\):**  
   The number 81 is equal to \( 3^4 \), so:
   \[
   81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27
   \]

3. **Simplify \( \left( y^{16} \right)^{\frac{3}{4}} \):**  
   Use the power of a power rule \( (x^m)^n = x^{m \cdot n} \):
   \[
   \left( y^{16} \right)^{\frac{3}{4}} = y^{16 \cdot \frac{3}{4}} = y^{12}
   \]

### Final answer:
\[
27 y^{12}
\]

Would you like any additional details or further clarification?

Here are some related questions for practice:

1. How would you simplify \( (16 x^8)^{\frac{1}{2}} \)?
2. What is the result of \( (64 a^4)^{\frac{5}{6}} \)?
3. How does the power of a product rule apply to fractional exponents?
4. What happens when you raise a number to a negative fractional power?
5. Can you generalize how to simplify expressions like \( (a b^m)^n \)?

**Tip:** When simplifying expressions with exponents, always break them down using basic exponent rules like distributing powers over products and using the power of a power rule.

This problem demonstrates how to simplify (81y^16)^(3/4) by applying exponent rules, including distributing powers over products and simplifying fractional exponents. Suitable for students in Grades 9-11, covering key algebraic concepts.

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