We are tasked with simplifying the expression:

$\frac{(3x^{-9})^3}{x^8 - 4}$

Step 1: Simplify the numerator

First, we need to simplify the numerator $(3x^{-9})^3$.

Apply the power rule for exponents $(a^m)^n = a^{m \cdot n}$:

$(3x^{-9})^3 = 3^3 \cdot (x^{-9})^3 = 27x^{-27}$

So the expression becomes:

$\frac{27x^{-27}}{x^8 - 4}$

Step 2: Final simplified form

At this point, the expression is simplified as much as possible:

$\frac{27x^{-27}}{x^8 - 4}$

This is the simplest form of the expression. There is no further simplification between the numerator and the denominator unless we factor the denominator, but $x^8 - 4$ doesn't factor nicely.

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Related Questions:

1. How do exponent rules apply when simplifying terms like $x^{-9}$?
2. Can the denominator $x^8 - 4$ be factored further?
3. What are some common properties of negative exponents?
4. How would this expression change if $x = 0$?
5. How would the simplification differ if $x$ were a specific number?

Tip:

Remember, when dealing with negative exponents, the expression can be rewritten as a fraction (i.e., $x^{-n} = \frac{1}{x^n}$).