We aim to simplify the given expression step-by-step, rewriting it with positive exponents.

Given Expression:

$\left( \frac{p^{-1/5} q^{-3/2}}{4^{-1} p^{-3} q^{-1/3}} \right)^{-3}$

Step 1: Simplify the fraction inside the parentheses

Group the terms in the numerator and denominator:

• Numerator: $p^{-1/5} q^{-3/2}$
• Denominator: $4^{-1} p^{-3} q^{-1/3}$

Combine powers of $p$, $q$, and constants: $\frac{p^{-1/5} q^{-3/2}}{4^{-1} p^{-3} q^{-1/3}} = 4^{1} \cdot p^{-1/5 - (-3)} \cdot q^{-3/2 - (-1/3)}$

Simplify the exponents:

1. For $p$: $-1/5 - (-3) = -1/5 + 3 = 14/5$
2. For $q$: $-3/2 - (-1/3) = -3/2 + 1/3 = -9/6 + 2/6 = -7/6$

So the expression becomes: $4 \cdot p^{14/5} \cdot q^{-7/6}$

Step 2: Apply the outer exponent $(-3)$

Raise each term to the power of $-3$: $\left( 4 \cdot p^{14/5} \cdot q^{-7/6} \right)^{-3} = 4^{-3} \cdot \left(p^{14/5}\right)^{-3} \cdot \left(q^{-7/6}\right)^{-3}$

Simplify each term:

1. $4^{-3} = 1/4^3 = 1/64$
2. $\left(p^{14/5}\right)^{-3} = p^{-(14/5) \cdot 3} = p^{-42/5}$
3. $\left(q^{-7/6}\right)^{-3} = q^{(-7/6) \cdot (-3)} = q^{21/6} = q^{7/2}$

Combine the results: $\frac{1}{64} \cdot p^{-42/5} \cdot q^{7/2}$

Step 3: Write with positive exponents

Rewrite $p^{-42/5}$ as $1/p^{42/5}$. The final simplified expression is: $\frac{q^{7/2}}{64 \cdot p^{42/5}}$

Final Answer:

$\boxed{\frac{q^{7/2}}{64p^{42/5}}}$

Would you like me to elaborate on any part of the solution?

---

Related Questions:

1. How do you simplify expressions involving negative exponents?
2. What are the steps to combine like terms in fractional exponents?
3. Why do we rewrite expressions with positive exponents?
4. How do we handle constants with negative powers, such as $4^{-1}$?
5. Can you provide another example of a similar simplification?

Tip:

Always simplify exponents before applying outer powers to reduce computational errors.