To simplify the expression \(\frac{10x^{-2}y^6}{-5x^{-5}y^{-2}}\), follow these steps:

1. **Simplify the coefficients**:
   \[
   \frac{10}{-5} = -2
   \]

2. **Apply the laws of exponents for \(x\) and \(y\)**:

   For \(x\):
   \[
   \frac{x^{-2}}{x^{-5}} = x^{-2 - (-5)} = x^{-2 + 5} = x^3
   \]

   For \(y\):
   \[
   \frac{y^6}{y^{-2}} = y^{6 - (-2)} = y^{6 + 2} = y^8
   \]

3. **Combine the results**:
   \[
   -2x^3y^8
   \]

The simplified expression is \(-2x^3y^8\).

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

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Here are some related questions you might find interesting:

1. How would the expression change if the exponents on \(x\) and \(y\) were different?
2. How can the laws of exponents be applied to simplify expressions with negative exponents?
3. What happens if the expression has fractional exponents instead of integers?
4. How does the sign of the coefficients impact the final simplified expression?
5. What are some common mistakes to avoid when simplifying expressions with multiple variables and exponents?

**Tip:** Remember that subtracting a negative exponent is equivalent to adding its positive value. This can simplify the process when dealing with complex expressions.

Learn how to simplify the expression 10x^-2 y^6 / -5x^-5 y^-2 step-by-step using the laws of exponents. This problem covers exponents, algebraic simplification, and negative coefficients. Suitable for students in Grades 7-9.

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