The expression you've provided is as follows:

$\left( \frac{a^2 b^{-\frac{1}{3}}}{a^{-\frac{2}{3}} b^{\frac{3}{4}}} \right)^3$

Steps to simplify:

1. Simplify the expression inside the parentheses: $\frac{a^2 b^{-\frac{1}{3}}}{a^{-\frac{2}{3}} b^{\frac{3}{4}}}$

   • For the powers of $a$: $a^2 \div a^{-\frac{2}{3}} = a^{2 - \left(-\frac{2}{3}\right)} = a^{2 + \frac{2}{3}} = a^{\frac{8}{3}}$
   • For the powers of $b$: $b^{-\frac{1}{3}} \div b^{\frac{3}{4}} = b^{-\frac{1}{3} - \frac{3}{4}} = b^{-\frac{1}{3} - \frac{9}{12}} = b^{-\frac{4}{12} - \frac{9}{12}} = b^{-\frac{13}{12}}$

   So the expression becomes: $a^{\frac{8}{3}} b^{-\frac{13}{12}}$

2. Apply the cube exponent to the entire expression: $\left( a^{\frac{8}{3}} b^{-\frac{13}{12}} \right)^3$

   • For $a$: $\left( a^{\frac{8}{3}} \right)^3 = a^{\frac{8}{3} \times 3} = a^8$
   • For $b$: $\left( b^{-\frac{13}{12}} \right)^3 = b^{-\frac{13}{12} \times 3} = b^{-\frac{39}{12}} = b^{-\frac{13}{4}}$

Thus, the final simplified expression is:

$a^8 b^{-\frac{13}{4}}$

Would you like any more details or explanations?

Related Questions:

1. How do you simplify expressions with exponents involving division?
2. What are the rules for multiplying and dividing powers of the same base?
3. How do you simplify fractional exponents?
4. What happens when a power is raised to another power?
5. How can negative exponents be interpreted in terms of fractions?

Tip:

When simplifying exponents, always apply the rules of exponents carefully (addition for multiplication, subtraction for division, and multiplication for powers of powers).