Let's simplify the given expression step by step. The expression is:

\[
\frac{(6 - 2a^2)^3}{12^3a^3} - 2
\]

### Step 1: Expand and simplify the numerator
First, simplify the numerator \((6 - 2a^2)^3\). Expanding it fully is complicated, but we can leave it in factored form for now.

\[
(6 - 2a^2)^3
\]

### Step 2: Simplify the denominator
The denominator is \(12^3a^3\). Calculate \(12^3\):

\[
12^3 = 12 \times 12 \times 12 = 1728
\]

Thus, the denominator becomes:

\[
1728a^3
\]

### Step 3: Write the expression as:

\[
\frac{(6 - 2a^2)^3}{1728a^3} - 2
\]

This is the simplified form of the expression, where \((6 - 2a^2)^3\) is left as it is in the numerator, and the denominator is simplified to \(1728a^3\). Subtract 2 at the end as indicated.

Do you have any questions or need further details? Here are some related questions you might find interesting:

1. How do you expand a binomial raised to a power, like \((6 - 2a^2)^3\)?
2. What are the properties of exponents used in simplifying powers?
3. How do you simplify expressions involving variables in both the numerator and denominator?
4. How do you handle negative exponents in algebraic expressions?
5. What are some common mistakes to avoid when simplifying algebraic fractions?

**Tip:** Always simplify expressions involving powers step by step, especially when dealing with large numbers or complex terms.

Simplify the expression (6 - 2a²)³ ÷ (12³a³) - 2 with a step-by-step breakdown. Learn about exponents, algebraic fractions, and polynomial simplifications. Perfect for high school students studying advanced algebra.

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