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Let's break down the problem step by step.

Given expression:
\[
\frac{(6x^5y^3)^2}{3x^2y^7 \times 4xy^{-3}}
\]

1. **Expand the Numerator:**
   \[
   (6x^5y^3)^2 = 6^2 \cdot (x^5)^2 \cdot (y^3)^2
   \]
   \[
   = 36x^{10}y^6
   \]

2. **Expand the Denominator:**
   \[
   3x^2y^7 \times 4xy^{-3} = (3 \times 4) \cdot (x^2 \times x) \cdot (y^7 \times y^{-3})
   \]
   \[
   = 12x^3y^4
   \]

3. **Combine and Simplify:**
   \[
   \frac{36x^{10}y^6}{12x^3y^4}
   \]

   Divide coefficients:
   \[
   \frac{36}{12} = 3
   \]

   Apply the quotient rule for exponents:
   \[
   x^{10} \div x^3 = x^{10-3} = x^7
   \]
   \[
   y^6 \div y^4 = y^{6-4} = y^2
   \]

   So, the simplified form is:
   \[
   3x^7y^2
   \]

Therefore, the expression in the form \( ax^b y^c \) is \( 3x^7y^2 \), where \( a = 3 \), \( b = 7 \), and \( c = 2 \).

Would you like a detailed explanation of any part? Here are some related questions to expand your understanding:

1. How do you simplify expressions with multiple variables?
2. What are the rules for multiplying and dividing exponents?
3. How do you handle negative exponents in an expression?
4. How can you verify the final simplified expression is correct?
5. What happens if one of the exponents is zero?

**Tip:** Remember that when simplifying algebraic fractions, always perform operations on the coefficients and variables separately for clarity.

Write (6x^5 y^3)^2 / 3x^2 y^7 × 4xy^-3 in the form ax^b y^c where a, b, and c are integers.

Learn how to simplify an algebraic expression with multiple exponents and variables in the form ax^b y^c. This example explores the laws of exponents and algebraic simplifications, suitable for students in Grades 9-12.

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