To solve the equation [tex]\(64^{3x} = 512^{2x + 12}\)[/tex], we'll start by expressing 64 and 512 as powers of a common base. Both 64 and 512 can be written as powers of 2:
[tex]\[
64 = 2^6
\][/tex]
[tex]\[
512 = 2^9
\][/tex]
Now we can rewrite the given equation using these facts:
[tex]\[
(2^6)^{3x} = (2^9)^{2x + 12}
\][/tex]
Using the power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[
2^{18x} = 2^{9(2x + 12)}
\][/tex]
Simplify the exponent on the right side:
[tex]\[
2^{18x} = 2^{18x + 108}
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
18x = 18x + 108
\][/tex]
Subtract [tex]\(18x\)[/tex] from both sides:
[tex]\[
0 = 108
\][/tex]
This is a false statement, which means there are no values of [tex]\(x\)[/tex] that satisfy the original equation. Thus, the solution is:
[tex]\[
\boxed{\text{no solution}}
\][/tex]
