Let's break down and simplify the given expression:

$\left(\frac{X^{12}}{81}\right)^{-\frac{4}{3}} \times \left(\frac{-x^9}{27}\right)^{-\frac{2}{3}} \times \left(\frac{-x^{-\frac{4}{5}}}{4}\right)^{-1}$

Step 1: Simplify each term.

First term:

$\left(\frac{X^{12}}{81}\right)^{-\frac{4}{3}} = \frac{X^{12 \times \left(-\frac{4}{3}\right)}}{81^{-\frac{4}{3}}} = \frac{X^{-16}}{81^{-\frac{4}{3}}}$

Now, simplify $81^{-\frac{4}{3}}$. We know $81 = 3^4$, so: $81^{-\frac{4}{3}} = \left(3^4\right)^{-\frac{4}{3}} = 3^{-16/3}$ Thus, the first term simplifies to: $\frac{X^{-16}}{3^{-\frac{16}{3}}} = X^{-16} \times 3^{\frac{16}{3}}$

Second term:

$\left(\frac{-x^9}{27}\right)^{-\frac{2}{3}} = \frac{\left(-x^9\right)^{-\frac{2}{3}}}{27^{-\frac{2}{3}}}$

Now simplify each part:

• For the numerator: $\left(-x^9\right)^{-\frac{2}{3}} = (-1)^{-\frac{2}{3}} x^{9 \times \left(-\frac{2}{3}\right)} = (-1)^{-\frac{2}{3}} x^{-6}$
• For the denominator: $27 = 3^3$, so $27^{-\frac{2}{3}} = (3^3)^{-\frac{2}{3}} = 3^{-2}$

Thus, the second term becomes: $\frac{(-1)^{-\frac{2}{3}} x^{-6}}{3^{-2}} = (-1)^{-\frac{2}{3}} x^{-6} \times 3^2$

Third term:

$\left(\frac{-x^{-\frac{4}{5}}}{4}\right)^{-1} = \frac{(-x^{-\frac{4}{5}})^{-1}}{4^{-1}}$

Now simplify each part:

• For the numerator: $(-x^{-\frac{4}{5}})^{-1} = (-1)^{-1} x^{\frac{4}{5}} = -x^{\frac{4}{5}}$
• For the denominator: $4^{-1} = \frac{1}{4}$

Thus, the third term becomes: $-4x^{\frac{4}{5}}$

Step 2: Combine all terms.

Now multiply all the simplified expressions: $\left(X^{-16} \times 3^{\frac{16}{3}}\right) \times \left((-1)^{-\frac{2}{3}} x^{-6} \times 3^2\right) \times \left(-4 x^{\frac{4}{5}}\right)$

Combine powers of $x$ and constants:

• For $x$: $X^{-16}$ is separate from $x$, and for $x$, we have powers: $-6 + \frac{4}{5}$.
• For constants: Combine powers of $3$ and other numbers.

Let me simplify the powers and the constants further.

After simplifying the given expression, the result is:

$-12616.80 \cdot X^{-16} \cdot x^{0.8}$

This simplifies further to: $-12616.80 \cdot X^{-16} \cdot x^{\frac{4}{5}}$

Thus, the final simplified expression is:

$-12616.80 \cdot X^{-16} \cdot x^{\frac{4}{5}}$

Would you like more details or have any questions?

Here are some related questions you could explore:

1. What is the meaning of fractional exponents like $x^{\frac{4}{5}}$?
2. How do negative exponents affect an expression?
3. What is the process to simplify terms with the same base but different powers?
4. How do constants behave during exponentiation when fractional powers are involved?
5. Can this result be interpreted geometrically or in real-world applications?

Tip: Always handle negative exponents carefully, as they often represent reciprocal operations.