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Let's break down the mathematical expression provided:

The given expression is:
\[
M(x, y) = (x^3y^2)^5 (x^3y^2)^5 \dots (x^3y^2)^5
\]

This sequence repeats 30 times. So, the entire expression can be written as:
\[
M(x, y) = \left[(x^3y^2)^5\right]^{30}
\]

Now, let's simplify this step-by-step:

1. **Simplify the Inner Term:** 
   \[
   (x^3y^2)^5 = x^{3 \times 5} y^{2 \times 5} = x^{15} y^{10}
   \]

2. **Expression with 30 repetitions:** 
   Since this term repeats 30 times, we can combine them using the property of exponents:
   \[
   \left(x^{15} y^{10}\right)^{30} = x^{15 \times 30} y^{10 \times 30} = x^{450} y^{300}
   \]

Therefore, the final simplified expression for \( M(x, y) \) is:
\[
M(x, y) = x^{450} y^{300}
\]

This is the generalized answer (GA) that the problem is asking for.

Would you like further details on this solution or have any questions?

Here are 5 related questions to deepen your understanding:
1. What properties of exponents were used in simplifying this expression?
2. How would the expression change if the repetition was different, say 20 times instead of 30?
3. What would happen if the powers inside the parentheses were different (e.g., \(x^2 y^3\))?
4. How would the simplification process differ if the terms were added instead of multiplied?
5. How would you represent \( M(x, y) \) if it was defined as a product of different terms rather than repeated identical terms?

**Tip:** When dealing with repeated multiplication of identical terms, use the properties of exponents to simplify the expression efficiently.

M(x, y) = (x^3y^2)^5 (x^3y^2)^5 … (x^3y^2)^5 repeated 30 times. Find GA.

Learn how to simplify the exponential expression (x^3y^2)^5 repeated 30 times. This problem involves the properties of exponents and exponentiation rules, suitable for students in Grades 10-12.

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