To solve the equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex], we begin by expressing the bases (64 and 512) as powers of 2.
1. [tex]\(64 = 2^6\)[/tex], so [tex]\(64^{3x} = (2^6)^{3x} = 2^{18x}\)[/tex].
2. [tex]\(512 = 2^9\)[/tex], so [tex]\(512^{2x + 12} = (2^9)^{2x + 12} = 2^{9(2x + 12)}\)[/tex].
Now, the original equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex] becomes:
[tex]\[2^{18x} = 2^{9(2x + 12)}\][/tex]
Since the bases are both 2, we can set the exponents equal to each other:
[tex]\[18x = 9(2x + 12)\][/tex]
Next, distribute the 9 on the right-hand side:
[tex]\[18x = 18x + 108\][/tex]
Subtract [tex]\(18x\)[/tex] from both sides:
[tex]\[18x - 18x = 108\][/tex]
[tex]\[0 = 108\][/tex]
This results in a contradiction, as [tex]\(0\)[/tex] cannot equal [tex]\(108\)[/tex]. Therefore, there is no value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex].
As a result, the correct answer is:
[tex]\[ \boxed{\text{no solution}} \][/tex]
