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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]
[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]
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