An exponential function in the form $y=64(b)^{x}$ contains the points $(0,64)$ and $(3,1)$. What is the value of $b$ ?
Do not include " $b=$ " in your answer.
Provide your answer below:
Nine 9
(i)
Answer
Final Answer: \(\boxed{\frac{1}{4}}\)
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: \(\boxed{\frac{1}{4}}\)
