To simplify the expression \((2y^5)^4 \cdot (3y^2)^{-3}\), follow these steps:

### Step 1: Simplify each term individually.

1. **Simplify \((2y^5)^4\):**
   - Apply the exponent 4 to both the constant 2 and the variable \(y^5\).
   \[
   (2^4) \cdot (y^5)^4 = 16y^{20}
   \]
   So, \((2y^5)^4\) simplifies to \(16y^{20}\).

2. **Simplify \((3y^2)^{-3}\):**
   - Apply the exponent \(-3\) to both the constant 3 and the variable \(y^2\).
   \[
   (3^{-3}) \cdot (y^2)^{-3} = \frac{1}{27} \cdot y^{-6} = \frac{y^{-6}}{27}
   \]

### Step 2: Multiply the simplified expressions.

Now multiply \(16y^{20}\) by \(\frac{y^{-6}}{27}\):
\[
\frac{16y^{20} \cdot y^{-6}}{27}
\]

### Step 3: Combine the exponents of \(y\).

Using the property \(y^a \cdot y^b = y^{a+b}\), add the exponents of \(y^{20}\) and \(y^{-6}\):
\[
y^{20} \cdot y^{-6} = y^{14}
\]

So, the expression simplifies to:
\[
\frac{16y^{14}}{27}
\]

### Final Answer:
\[
\frac{16y^{14}}{27}
\]

Would you like any further explanations or have any questions about the steps?

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Here are 8 follow-up questions:

1. How would you simplify \((4x^3)^2 \cdot (2x^{-1})^3\)?
2. Can you solve \((5z^4)^{-2} \cdot (3z^3)^3\)?
3. What happens when multiplying terms with different bases but the same exponents?
4. How would you simplify an expression with multiple negative exponents?
5. What is the effect of raising a product to an exponent?
6. How do you simplify \((6p^7)^2 \cdot (2p^{-4})^3\)?
7. What if the bases are fractions, such as \((\frac{1}{2}y^3)^4\)?
8. Can you demonstrate how to simplify \((9q^5)^{-1} \cdot (4q^2)^3\)?

**Tip:** When multiplying expressions, combine like terms by adding exponents for the same base.

Learn how to simplify the expression (2y^5)^4 * (3y^2)^(-3) with a detailed step-by-step solution. This problem covers exponentiation and multiplication of exponents, including negative exponents. Suitable for students in Grades 7-9.

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