The expression \(\left( \frac{p^3}{2} \right)^{-2}\) can be simplified using the rules of exponents. Here's the step-by-step process:

1. Apply the exponent rule \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\):
   \[
   \left( \frac{p^3}{2} \right)^{-2} = \frac{\left( p^3 \right)^{-2}}{2^{-2}}
   \]

2. Simplify each part:
   - For the numerator: \(\left( p^3 \right)^{-2} = p^{3 \times (-2)} = p^{-6}\)
   - For the denominator: \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\)

   This gives:
   \[
   \frac{p^{-6}}{\frac{1}{4}} = p^{-6} \times 4
   \]

3. Finally, simplify the result:
   \[
   4p^{-6} = \frac{4}{p^6}
   \]

So, \(\left( \frac{p^3}{2} \right)^{-2} = \frac{4}{p^6}\).

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

Here are 5 related questions:
1. How would you simplify \(\left( \frac{p^2}{3} \right)^{-3}\)?
2. What happens when you apply a positive exponent instead of a negative one?
3. Can you simplify \(\left( \frac{p^5}{4q^3} \right)^{-2}\)?
4. How does the rule \((ab)^n = a^n b^n\) help simplify expressions like \(\left( \frac{p^4}{6} \right)^3\)?
5. How would the answer change if \(p\) had a coefficient in front of it, like \(2p^3\)?

**Tip:** When dealing with negative exponents, remember that they indicate reciprocation of the base.

Learn how to simplify the expression (p^3/2)^-2 step by step, using exponent rules. This guide covers exponentiation of fractional expressions and negative exponents, suitable for students in Grades 8-10.

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