((3x^(-9))^(3))/((x^(8))-4) simplify the equation

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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Would you like to dive into more details or ask any specific questions?

### 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., \(x^{-n} = \frac{1}{x^n}\)).

This solution walks through the steps of simplifying the expression ((3x^(-9))^3) / ((x^(8))-4) using exponent rules. It covers algebraic concepts such as power rules and negative exponents, making it ideal for students in Grades 9-12.

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