Let's solve the equation step-by-step to find the value of [tex]\( x \)[/tex] for which the equation holds true.
We are given the equation:
[tex]\[
\frac{512^{x-2}}{\left(\frac{1}{64}\right)^{3x}} = 512
\][/tex]
First, let's simplify this expression. We know that:
[tex]\[
512 = 2^9 \quad \text{and} \quad \frac{1}{64} = 64^{-1} = (2^6)^{-1} = 2^{-6}
\][/tex]
Therefore, we can rewrite the equation using these exponents:
[tex]\[
\frac{(2^9)^{x-2}}{(2^{-6})^{3x}} = 2^9
\][/tex]
Simplify the exponents in the numerator and the denominator:
[tex]\[
\frac{2^{9(x-2)}}{2^{-18x}} = 2^9
\][/tex]
Combine the exponents using the rules of exponents:
[tex]\[
2^{9(x-2) - (-18x)} = 2^9
\][/tex]
Simplify the exponent in the numerator:
[tex]\[
2^{9(x-2) + 18x} = 2^9
\][/tex]
Combine like terms in the exponent:
[tex]\[
2^{9x - 18 + 18x} = 2^9
\][/tex]
[tex]\[
2^{27x - 18} = 2^9
\][/tex]
Since the bases are the same (both are 2), we can set the exponents equal to each other:
[tex]\[
27x - 18 = 9
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
27x - 18 = 9
\][/tex]
[tex]\[
27x = 27
\][/tex]
[tex]\[
x = 1
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = 1
\][/tex]
So, the answer is:
[tex]\[
\boxed{x = 1}
\][/tex]
