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To factor the expression \( 27x^6 - 48 \) completely, follow these steps:

1. **Find the Greatest Common Factor (GCF):**  
   The GCF of \( 27x^6 \) and \( 48 \) is \( 3 \). So, factor out \( 3 \):
   \[
   27x^6 - 48 = 3(9x^6 - 16)
   \]

2. **Recognize the Difference of Squares:**  
   The expression inside the parentheses, \( 9x^6 - 16 \), is a difference of squares because \( 9x^6 = (3x^3)^2 \) and \( 16 = 4^2 \):
   \[
   9x^6 - 16 = (3x^3)^2 - 4^2
   \]

   Using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \):
   \[
   9x^6 - 16 = (3x^3 - 4)(3x^3 + 4)
   \]

3. **Combine All Factors:**  
   Now, substitute back to get the completely factored form:
   \[
   27x^6 - 48 = 3(3x^3 - 4)(3x^3 + 4)
   \]

So, the expression \( 27x^6 - 48 \) factors completely to:
\[
3(3x^3 - 4)(3x^3 + 4)
\]

Would you like further explanation or details on any part of this process? 

Here are some related questions to consider:
1. How do you factor a polynomial that is not a difference of squares?
2. What is the significance of factoring in solving equations?
3. How can you identify the greatest common factor in more complex expressions?
4. What is the difference between factoring and expanding a polynomial?
5. How does the difference of squares formula apply to more complex algebraic expressions?

**Tip:** Always look for the greatest common factor (GCF) first when factoring any polynomial to simplify the process.

Learn how to factor the algebraic expression 27x^6 - 48 completely by finding the greatest common factor (GCF) and applying the difference of squares. Step-by-step explanation provided, suitable for students in Grades 9-11.

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