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To simplify the expression:

\[
\frac{4x^{10}y^{-3}}{12x^{-2}y}
\]

we'll follow these steps:

1. **Simplify the coefficients:**
   \[
   \frac{4}{12} = \frac{1}{3}
   \]

2. **Simplify the powers of \(x\):**
   \[
   \frac{x^{10}}{x^{-2}} = x^{10 - (-2)} = x^{10 + 2} = x^{12}
   \]

3. **Simplify the powers of \(y\):**
   \[
   \frac{y^{-3}}{y^1} = y^{-3 - 1} = y^{-4}
   \]

Putting everything together, the simplified expression is:

\[
\frac{x^{12}}{3y^4}
\]

Thus, the correct answer is **B**.

---

Do you want more details or have any questions about this? Here are five related questions to further your understanding:

1. How do we handle negative exponents when simplifying algebraic expressions?
2. Why does the exponent rule \(\frac{a^m}{a^n} = a^{m-n}\) work?
3. What happens if there are variables with the same exponent in both the numerator and denominator?
4. How would you simplify \(\frac{2x^{-4}y^2}{5x^6y^{-3}}\)?
5. What role do coefficients play in simplifying algebraic fractions?

**Tip**: Always simplify the numerical coefficients before applying exponent rules to the variables!

Simplify the expression \( \frac{4x^{10}y^{-3}}{12x^{-2}y} \)

Learn how to simplify the expression \( \frac{4x^{10}y^{-3}}{12x^{-2}y} \). This problem covers key algebraic concepts such as simplifying rational expressions and exponent rules. Suitable for students in Grades 9-11.

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