To factorize the expression [tex]\( 216 x^{12} - 64 \)[/tex] properly, we need to follow a structured approach.
First, observe that we can factor out a common term from the given expression:
\- [tex]\( 216 x^{12} - 64 \)[/tex]
There is a common factor of 8, so we start by factoring that out:
[tex]\[ 216 x^{12} - 64 = 8 \cdot (27 x^{12} - 8) \][/tex]
Next, we focus on factoring the expression inside the parentheses [tex]\( 27 x^{12} - 8 \)[/tex]. Notice that this expression takes the form of a difference of cubes because [tex]\( 27x^{12} = (3x^4)^3 \)[/tex] and [tex]\( 8 = 2^3 \)[/tex]:
[tex]\[ 27 x^{12} - 8 = (3x^4)^3 - 2^3 \][/tex]
Using the difference of cubes formula, [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex], where [tex]\( a = 3x^4 \)[/tex] and [tex]\( b = 2 \)[/tex]:
[tex]\[ (3x^4)^3 - 2^3 = (3x^4 - 2)\left((3x^4)^2 + (3x^4)(2) + 2^2\right) \][/tex]
Compute each part within the parentheses:
[tex]\[ (3x^4)^2 = 9x^8 \][/tex]
[tex]\[ (3x^4)(2) = 6x^4 \][/tex]
[tex]\[ 2^2 = 4 \][/tex]
Now, combine these parts within the parentheses:
[tex]\[ (3x^4)^3 - 2^3 = (3x^4 - 2)(9x^8 + 6x^4 + 4) \][/tex]
Therefore, combining the factored term of 8 that we initially factored out, we get:
[tex]\[ 216 x^{12} - 64 = 8 \cdot (3x^4 - 2)(9x^8 + 6x^4 + 4) \][/tex]
So, the factorized form of [tex]\( 216 x^{12} - 64 \)[/tex] is:
[tex]\[ 8(3x^4 - 2)(9x^8 + 6x^4 + 4) \][/tex]
Given the answer choices:
1. [tex]\( (6 x^3-4)(36 x^6+24 x^3+16) \)[/tex]
2. [tex]\( (6 x^3-4)(36 x^9+24 x^3+16) \)[/tex]
3. [tex]\( (6 x^4-4)(36 x^8+24 x^4+16) \)[/tex]
4. [tex]\( (6 x^4-4)(36 x^{12}+24 x^4+16) \)[/tex]
None of these match with 8 being factored out. However, when matching the inside factored expressions correctly, [tex]\( 8(3 x^4 - 2)(9 x^8 + 6 x^4 + 4) \)[/tex] is equivalent to:
[tex]\[ 8 \cdot (3 x^4 - 2) \cdot (9 x^8 + 6 x^4 + 4) \][/tex]
which simplifies to
[tex]\[ (3 x^4 - 2) \cdot (9 x^8 + 6 x^4 + 4 \][/tex]
Since none of the answer choices provided match exactly, the closest starting point based on matching terms should be revised. The correct answer considering the originally unreasoned factorization in interpretation and provided true computations would stand formally clarified yet among options illustrated incorrect derivatively.
So, the correct derived form when factoring classically through provision manner is:
[tex]\[ \boxed{8(3x^4 - 2)(9x^8 + 6x^4 + 4)} \][/tex] displaying comprehended understanding aligned with true illustrated computations.
Hence, the direct printed choice does not perfectly adhere to proven forms factor clarity stated comparatives independently verified for correctness outside presented choices.
