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The given expression is:

\[
\frac{4n^6}{2n^{-2}}
\]

To simplify this, follow these steps:

1. **Simplify the numerical coefficients**: \(\frac{4}{2} = 2\).

2. **Apply the properties of exponents**: Use the rule \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the powers of \(n\):
   
   \[
   \frac{n^6}{n^{-2}} = n^{6 - (-2)} = n^{6 + 2} = n^8.
   \]

3. **Combine the results**: Now multiply the simplified coefficient with the simplified power of \(n\):

   \[
   2n^8
   \]

So, the simplified expression is:

\[
2n^8
\]

Would you like any further explanation or have any questions?

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Here are 5 related questions to explore:

1. How would the expression change if the denominator had a positive exponent?
2. What is the general rule for dividing powers with the same base?
3. How do you simplify an expression with variables in both the numerator and denominator?
4. What if the coefficient in the numerator was not divisible by the denominator? How would you handle that?
5. What happens if one of the exponents is a fraction?

**Tip**: Always apply exponent rules carefully when simplifying expressions. Check both the coefficient and the exponent terms.

In this problem, we simplify the algebraic expression 4n^6 / 2n^-2. Learn how to apply exponent rules and divide terms with the same base to get the simplified result. Suitable for students in Grades 8-10.

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