We are given the equation $4^{5-9x} = \frac{1}{8^{x-2}}$ and need to solve for $x$.

1. Problem Description

We are given the equation $4^{5-9x} = \frac{1}{8^{x-2}}$ and need to solve for $x$.

2. Solution Steps

First, rewrite both sides of the equation using base 2:

$4 = 2^2$ and $8 = 2^3$.

Substitute these into the equation:

$(2^2)^{5-9x} = \frac{1}{(2^3)^{x-2}}$

Using the power of a power rule, $(a^m)^n = a^{mn}$, we get:

$2^{2(5-9x)} = \frac{1}{2^{3(x-2)}}$

Using the property $\frac{1}{a^n} = a^{-n}$, we have:

$2^{2(5-9x)} = 2^{-3(x-2)}$

Since the bases are equal, we can equate the exponents:

$2(5-9x) = -3(x-2)$

Expand both sides:

$10 - 18x = -3x + 6$

Add $18x$ to both sides:

$10 = 15x + 6$

Subtract 6 from both sides:

$4 = 15x$

Divide by 15:

$x = \frac{4}{15}$

3. Final Answer

$x = \frac{4}{15}$