Find the value of \( x \) if \( \left(\frac{1}{8}\right)^{2 x-9}=4^{-\frac{3}{2} x} \)

To solve the equation \( \left(\frac{1}{8}\right)^{2 x-9}=4^{-\frac{3}{2} x} \), we start by expressing both sides with the same base. First, rewrite \( \frac{1}{8} \) and \( 4 \) in terms of base \( 2 \): \[ \frac{1}{8} = 2^{-3} \quad \text{and} \quad 4 = 2^2. \] Now, we can rewrite the equation: \[ \left(2^{-3}\right)^{2x-9} = (2^2)^{-\frac{3}{2} x}. \] Applying the power of a power property, we simplify both sides: \[ 2^{-3(2x-9)} = 2^{2 \left(-\frac{3}{2} x\right)}. \] This simplifies to: \[ 2^{-6x + 27} = 2^{-3x}. \] Since the bases are the same, we can set the exponents equal to each other: \[ -6x + 27 = -3x. \] Next, solve for \( x \): \[ 27 = -3x + 6x, \] \[ 27 = 3x, \] \[ x = \frac{27}{3} = 9. \] Thus, the value of \( x \) is \[ \boxed{9}. \]