Step 1 :Given an exponential function in the form \(y=64(b)^{x}\) and the points (0,64) and (3,1).
Step 2 :Substitute the given points into the function to form two equations. The point (0,64) gives us the equation \(64 = 64 * b^0\). The point (3,1) gives us the equation \(1 = 64 * b^3\).
Step 3 :Solving these equations, we get three possible values for \(b\).
Step 4 :However, since \(b\) is in the base of an exponential function, it must be a positive real number. Therefore, the only valid solution is \(b = \frac{1}{4}\).
Step 5 :Final Answer: \(\boxed{\frac{1}{4}}\)