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The question provided asks to simplify the expression:

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

in the form \( ax^by^c \), where \(a\), \(b\), and \(c\) are integers.

### Step-by-step solution:

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

2. **Expand the denominator:**
   \[
   3x^2y^7 \times 4xy^{-3} = 3 \times 4 \times x^2 \times x^1 \times y^7 \times y^{-3} = 12x^3y^4
   \]

3. **Divide the terms:**
   \[
   \frac{36x^{10}y^6}{12x^3y^4} = \frac{36}{12} \times \frac{x^{10}}{x^3} \times \frac{y^6}{y^4}
   \]
   Simplifying each factor:
   \[
   \frac{36}{12} = 3, \quad \frac{x^{10}}{x^3} = x^{10-3} = x^7, \quad \frac{y^6}{y^4} = y^{6-4} = y^2
   \]

4. **Final result:**
   \[
   3x^7y^2
   \]

Thus, the expression is simplified to \( 3x^7y^2 \).

---

Would you like further details or explanations on any step? Here are some related questions:

1. How do you expand exponents like \((x^5)^2\)?
2. What are the rules for simplifying fractions with variables?
3. How do you combine powers when multiplying terms with the same base?
4. Why does dividing powers result in subtraction of exponents?
5. What does the expression \( ax^by^c \) represent mathematically?

**Tip:** When simplifying algebraic expressions, always handle each variable separately using exponent rules.

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

Learn how to simplify the algebraic expression (6x^5y^3)^2 / (3x^2y^7 × 4xy^-3) into the form ax^by^c. This problem covers the laws of exponents, including multiplying and dividing powers, and is suitable for high school students in Grades 9-12.

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