If f(x) is an exponential function of the form of y=ab^{x} where f(4)=16 and f(4.5)=54, then find the value of f(5), to the nearest hundredth.

To solve for \( f(5) \), we start with the given form of the function \( f(x) = a b^{x} \) and the provided values \( f(4) = 16 \) and \( f(4.5) = 54 \). Setting up the equations with the known values: 1. From \( f(4) = 16 \): \[ a b^4 = 16 \quad (1) \] 2. From \( f(4.5) = 54 \): \[ a b^{4.5} = 54 \quad (2) \] Next, we can express \( b^{4.5} \) in terms of \( b^4 \): \[ b^{4.5} = b^4 \cdot b^{0.5} \] Substituting this into equation (2): \[ a b^4 \cdot b^{0.5} = 54 \] Now, replacing \( a b^4 \) from equation (1): \[ 16 \cdot b^{0.5} = 54 \] Solving for \( b^{0.5} \): \[ b^{0.5} = \frac{54}{16} = 3.375 \] To find \( b \), we square both sides: \[ b = (3.375)^2 = 11.390625 \] Now, substitute \( b \) back into equation (1) to find \( a \): \[ a (11.390625)^4 = 16 \] Calculating \( (11.390625)^4 \): \[ (11.390625)^4 \approx 16484.96449 \] Then, \[ a \cdot 16484.96449 = 16 \implies a = \frac{16}{16484.96449} \approx 0.000969 \] Now we can express \( f(x) \) fully: \[ f(x) = 0.000969 \cdot (11.390625)^x \] Finally, we need to find \( f(5) \): \[ f(5) = 0.000969 \cdot (11.390625)^5 \] Calculating \( (11.390625)^5 \): \[ (11.390625)^5 \approx 187384.6237 \] Then, \[ f(5) \approx 0.000969 \cdot 187384.6237 \approx 181.539 \] Thus, the value of \( f(5) \) to the nearest hundredth is: \[ \boxed{181.54} \]