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The given expression is quite complex. To solve it, we'll first break it down and simplify both the numerator and the denominator step by step.

The expression is:

\[
\frac{\left( 27a^6b^{-9} \right)^{\frac{1}{3}} \left( \frac{a^{-2}b^5}{81a^3b^{-4}} \right)^{-\frac{1}{4}}}{\left( \frac{64a^{-8}b^{10}}{a^4b^{-2}} \right)^{\frac{1}{6}}}
\]

### Step 1: Simplify powers of numbers and variables in the numerator and denominator

#### Numerator
- First part: \( (27a^6b^{-9})^{\frac{1}{3}} \)
    - \( 27^{\frac{1}{3}} = 3 \)
    - \( (a^6)^{\frac{1}{3}} = a^2 \)
    - \( (b^{-9})^{\frac{1}{3}} = b^{-3} \)
    
    So the first part simplifies to: \( 3a^2b^{-3} \).

- Second part: \( \left( \frac{a^{-2}b^5}{81a^3b^{-4}} \right)^{-\frac{1}{4}} \)
    - Simplifying inside the fraction:
        - \( a^{-2} / a^3 = a^{-5} \)
        - \( b^5 / b^{-4} = b^9 \)
        - \( 81 = 3^4 \), so \( 81^{-\frac{1}{4}} = 3^{-1} = \frac{1}{3} \)
        
    So the second part simplifies to: \( \left( \frac{a^{-5}b^9}{81} \right)^{-\frac{1}{4}} = \frac{1}{3} a^{5/4} b^{-9/4} \).

#### Denominator
- \( \left( \frac{64a^{-8}b^{10}}{a^4b^{-2}} \right)^{\frac{1}{6}} \)  
  


Simplify the complex algebraic expression involving powers of numbers and variables, radicals, and fractions.

Learn how to simplify complex algebraic expressions with powers, radicals, and fractions. This problem focuses on the properties of exponents, simplifying fractions, and radical expressions, suitable for high school students in grades 9-12.

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