Simplify. Write your answer without negative exponents. Assume that no denominator equals zero.

x-6w7n/x5w-2n4

To simplify the expression \(\frac{x^{-6}w^{7}n}{x^{5}w^{-2}n^{4}}\), we follow these steps:

1. **Combine the like terms in the numerator and the denominator:**

   \[
   \frac{x^{-6}w^{7}n}{x^{5}w^{-2}n^{4}} = \frac{x^{-6}}{x^{5}} \cdot \frac{w^{7}}{w^{-2}} \cdot \frac{n}{n^{4}}
   \]

2. **Apply the quotient rule for exponents:**

   - For \(x\), subtract the exponent in the denominator from the exponent in the numerator:
     \[
     \frac{x^{-6}}{x^{5}} = x^{-6 - 5} = x^{-11}
     \]

   - For \(w\), subtract the exponent in the denominator from the exponent in the numerator:
     \[
     \frac{w^{7}}{w^{-2}} = w^{7 - (-2)} = w^{7 + 2} = w^{9}
     \]

   - For \(n\), subtract the exponent in the denominator from the exponent in the numerator:
     \[
     \frac{n}{n^{4}} = n^{1 - 4} = n^{-3}
     \]

3. **Combine the results:**
   \[
   \frac{x^{-6}w^{7}n}{x^{5}w^{-2}n^{4}} = x^{-11}w^{9}n^{-3}
   \]

4. **Rewrite with positive exponents:**

   \[
   x^{-11}w^{9}n^{-3} = \frac{w^{9}}{x^{11}n^{3}}
   \]

So, the simplified expression is:

\[
\frac{w^{9}}{x^{11}n^{3}}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions:
1. How would you simplify \(\frac{x^{3}y^{-4}}{x^{-2}y^{5}}\)?
2. Can you simplify \(\frac{a^{-3}b^{2}}{a^{5}b^{-1}}\)?
3. What is the result of simplifying \(\frac{p^{-7}q^{4}}{p^{2}q^{-3}}\)?
4. How would you write \(\frac{m^{-2}n^{3}}{m^{5}n^{-4}}\) without negative exponents?
5. How do you simplify an expression like \(\frac{x^{-a}y^{b}}{x^{c}y^{-d}}\)?

**Tip:** When simplifying expressions, always remember to apply the rules of exponents carefully, particularly when dealing with negative exponents.

Learn how to simplify the fraction \(\frac{x^{-6}w^{7}n}{x^{5}w^{-2}n^{4}}\) without negative exponents using the quotient rule for exponents. This problem involves exponents and fraction simplification, suitable for Grades 7-9.

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