24. Solve for \( x: \sqrt{(x-2)^{-3}}=64 \)

To solve for \( x \) in the equation \( \sqrt{(x-2)^{-3}}=64 \), we can start by squaring both sides to eliminate the square root: \[ (x-2)^{-3} = 64^2 \] Calculating \( 64^2 \) gives us \( 4096 \): \[ (x-2)^{-3} = 4096 \] Next, we can rewrite this as: \[ \frac{1}{(x-2)^3} = 4096 \] To isolate \( (x-2)^3 \), we take the reciprocal of both sides: \[ (x-2)^3 = \frac{1}{4096} \] Recognizing that \( 4096 = 2^{12} \), we can express the equation as: \[ (x-2)^3 = 2^{-12} \] Taking the cube root of both sides gives: \[ x-2 = 2^{-4} \] Since \( 2^{-4} = \frac{1}{16} \), we can rewrite the equation as: \[ x-2 = \frac{1}{16} \] Now, solving for \( x \): \[ x = \frac{1}{16} + 2 = \frac{1}{16} + \frac{32}{16} = \frac{33}{16} \] Thus, the solution is: \[ \boxed{\frac{33}{16}} \]