To simplify the given expression \(\frac{(6y^3)^4}{2y^9}\), follow these steps:

1. **Simplify the numerator \((6y^3)^4\):**

   \[
   (6y^3)^4 = 6^4 \cdot (y^3)^4
   \]
   
   Calculate \(6^4\) and \((y^3)^4\):
   
   \[
   6^4 = 1296 \quad \text{and} \quad (y^3)^4 = y^{12}
   \]
   
   So, the numerator becomes:
   
   \[
   1296y^{12}
   \]

2. **Rewrite the entire expression:**

   \[
   \frac{1296y^{12}}{2y^9}
   \]

3. **Simplify the fraction by dividing the coefficients and using the properties of exponents:**

   - Divide the coefficients \(1296\) and \(2\):
     
     \[
     \frac{1296}{2} = 648
     \]
   
   - Subtract the exponents of \(y\) in the denominator from the numerator:

     \[
     y^{12} \div y^9 = y^{12-9} = y^3
     \]

4. **Combine these results:**

   \[
   648y^3
   \]

Thus, the simplified form of the expression is:

\[
\boxed{648y^3}
\]

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you apply the power of a power rule for exponents in other scenarios?
2. What is the general rule for simplifying expressions with different bases?
3. How do you handle expressions with negative exponents?
4. What are some techniques to simplify complex algebraic fractions?
5. How can these simplification techniques be applied to solve equations?

**Tip:** When working with exponents, always remember the power rules: \( (a^m)^n = a^{m \cdot n} \) and \( \frac{a^m}{a^n} = a^{m-n} \).

Learn how to simplify the expression ((6y^3)^4)/2y^9 step-by-step. Follow the rules of exponents to simplify the numerator and denominator, and understand the power of a power rule. Suitable for students in Grades 7-9 studying algebraic expressions.

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