The expression \( A^6 - \frac{1}{27} \) can be recognized as a difference of cubes. We can factor this using the difference of cubes formula, which is given by:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

To apply this formula, we first rewrite the expression as:

\[ A^6 - \left(\frac{1}{3}\right)^3 \]

Here, \( a = A^2 \) and \( b = \frac{1}{3} \).

Now, we use the formula:

1. **Identify \( a \) and \( b \)**:
   \[ a = A^2, \quad b = \frac{1}{3} \]

2. **Apply the difference of cubes formula**:
   \[ A^6 - \frac{1}{27} = \left(A^2\right)^3 - \left(\frac{1}{3}\right)^3 = \left(A^2 - \frac{1}{3}\right)\left((A^2)^2 + A^2\left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2\right) \]

Let's simplify the second part:

- The first term: \((A^2)^2 = A^4\)
- The second term: \(A^2\left(\frac{1}{3}\right) = \frac{A^2}{3}\)
- The third term: \(\left(\frac{1}{3}\right)^2 = \frac{1}{9}\)

So, the factorization becomes:

\[ A^6 - \frac{1}{27} = \left(A^2 - \frac{1}{3}\right)\left(A^4 + \frac{A^2}{3} + \frac{1}{9}\right) \]

This is the fully factored form.

Would you like more details or have any questions?

### Related Questions
1. How do you factor a sum of cubes?
2. Can you demonstrate how to expand a factored difference of cubes expression?
3. What are the real and complex roots of the equation \( A^6 - \frac{1}{27} = 0 \)?
4. How can you verify the factorization of \( A^6 - \frac{1}{27} \)?
5. What are the applications of factoring in solving equations?
6. How can this factoring technique be applied to more complex expressions?
7. What is the significance of the difference of cubes formula in algebra?
8. How can we graph the function \( A^6 - \frac{1}{27} \) and its factored form?

**Tip:** When working with expressions involving fractions, it's helpful to rewrite them in a way that makes the pattern (like a sum or difference of cubes) more recognizable.

Explore how to factor the expression A^6 - 1/27 using the difference of cubes formula. This detailed solution covers the application of algebraic concepts such as polynomials and factoring, with step-by-step explanations and examples. Suitable for high school students studying algebra and polynomial equations.

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