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The expression given is:

\[
\left( \frac{1}{3} x^4 y^7 \right)^2 \cdot \left( \frac{4}{2} x^2 y^5 \right)^2
\]

Let's solve this step-by-step.

### Step 1: Simplify Each Term Inside the Parentheses

1. For the first part, \(\left( \frac{1}{3} x^4 y^7 \right)^2\):
   - Apply the power rule \((a \cdot b)^n = a^n \cdot b^n\).
   - This becomes:
     \[
     \frac{1^2}{3^2} \cdot x^{4 \cdot 2} \cdot y^{7 \cdot 2} = \frac{1}{9} x^8 y^{14}
     \]

2. For the second part, \(\left( \frac{4}{2} x^2 y^5 \right)^2\):
   - Simplify inside the parentheses first:
     \[
     \frac{4}{2} = 2
     \]
   - Now we have \((2 x^2 y^5)^2\).
   - Applying the power rule again:
     \[
     2^2 \cdot x^{2 \cdot 2} \cdot y^{5 \cdot 2} = 4 x^4 y^{10}
     \]

### Step 2: Multiply the Results

Now we have:
\[
\frac{1}{9} x^8 y^{14} \cdot 4 x^4 y^{10}
\]

- Combine the constants:
  \[
  \frac{1}{9} \cdot 4 = \frac{4}{9}
  \]

- Combine the powers of \(x\):
  \[
  x^{8 + 4} = x^{12}
  \]

- Combine the powers of \(y\):
  \[
  y^{14 + 10} = y^{24}
  \]

### Final Answer

\[
\frac{4}{9} x^{12} y^{24}
\]

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Would you like a breakdown of any specific step? Here are some related questions:

1. How would this process change if there were different exponents?
2. What if there were negative exponents in the expression?
3. How do we handle fractional exponents in a similar expression?
4. How does simplifying change if we have terms without exponents?
5. What if the terms involved complex numbers?

**Tip:** Always simplify within parentheses before applying exponents outside.

(1/3 * x^4 * y^7)^2 * (4/2 * x^2 * y^5)^2

Explore step-by-step how to simplify the algebraic expression (1/3 * x^4 * y^7)^2 * (4/2 * x^2 * y^5)^2. This problem demonstrates exponent rules, multiplication of terms, and simplification techniques. Suitable for high school students, Grades 9-12.

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