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

$\frac{4}{9} x^{12} y^{24}$

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.