To solve the expression \(\frac{(2x^2 y^{-1})^7}{(4x^2)^3}\), we follow these steps:

### Step 1: Simplify the numerator and denominator separately.
- **Numerator:** \((2x^2 y^{-1})^7\)

  Apply the power to each factor inside the parentheses:
  \[
  (2^7)(x^{2 \cdot 7})(y^{-1 \cdot 7}) = 128x^{14}y^{-7}
  \]

- **Denominator:** \((4x^2)^3\)

  Again, apply the power to each factor:
  \[
  (4^3)(x^{2 \cdot 3}) = 64x^6
  \]

### Step 2: Combine the simplified numerator and denominator.
\[
\frac{128x^{14}y^{-7}}{64x^6}
\]

### Step 3: Simplify the fraction.
- Simplify the coefficients: \(\frac{128}{64} = 2\)
- Simplify the powers of \(x\): \(\frac{x^{14}}{x^6} = x^{14-6} = x^8\)

So, the expression becomes:
\[
2x^8 y^{-7}
\]

### Step 4: Express the answer in the simplest form.
- Since \(y^{-7} = \frac{1}{y^7}\), the final answer is:
\[
\frac{2x^8}{y^7}
\]

This is the simplified form of the given expression.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the solution change if the exponent in the numerator were different, say \((2x^2 y^{-1})^5\)?
2. Can you solve a similar problem with different variables, such as \(\frac{(3a^3 b^{-2})^4}{(9a^2)^2}\)?
3. What would be the effect of having a negative exponent in the denominator?
4. How would you simplify \(\frac{(5x^3 y^2)^{4}}{(25x y^{-1})^2}\)?
5. What are the steps to simplify expressions involving fractional exponents?

**Tip:** Always apply the exponent rules carefully, especially when dealing with negative exponents and coefficients, to avoid mistakes in simplification.

Learn how to simplify the expression (2x^2 y^-1)^7 / (4x^2)^3 with a step-by-step explanation. This problem covers exponent rules, simplifying expressions with variables, and fractional exponents. Suitable for students in Grades 7-9.

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