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The expression given is:

\[
\frac{45x^{-6}y^{-5}}{5x^{-7}y^{-1}}
\]

### Step-by-step simplification:

1. **Simplify the constants (numerical coefficients)**:
   \[
   \frac{45}{5} = 9
   \]
   So the expression becomes:
   \[
   9 \cdot \frac{x^{-6}y^{-5}}{x^{-7}y^{-1}}
   \]

2. **Simplify the powers of \(x\)**:
   When dividing powers of the same base, subtract the exponents:
   \[
   x^{-6} \div x^{-7} = x^{-6 - (-7)} = x^{1}
   \]
   Now the expression is:
   \[
   9 \cdot x^1 \cdot \frac{y^{-5}}{y^{-1}}
   \]

3. **Simplify the powers of \(y\)**:
   \[
   y^{-5} \div y^{-1} = y^{-5 - (-1)} = y^{-4}
   \]
   The expression is now:
   \[
   9 \cdot x \cdot y^{-4}
   \]

4. **Final result**:
   \[
   9x y^{-4}
   \]

This is the simplified form of the given expression.

Would you like more details on any step, or have additional questions?

### Relative Questions:
1. How do you handle division of exponents with negative powers?
2. What happens when you subtract negative exponents?
3. Can you explain how to simplify fractional expressions with variables?
4. How can you rewrite negative exponents in a positive form?
5. What are the rules for multiplying and dividing exponents?

**Tip:** When dividing terms with exponents, remember to subtract the exponent of the denominator from the exponent of the numerator!

Simplify the expression: 45x^(-6) y^(-5) / 5x^(-7) y^(-1)

Learn how to simplify the expression 45x^(-6) y^(-5) / 5x^(-7) y^(-1) using the laws of exponents. This problem explains how to divide powers and simplify terms with negative exponents. Suitable for students in Grades 8-10.

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