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Given the points $ ( 2 , 3 ) $ and $ ( 5 , 54 ) $ are on an exponential model, find the equation for this model in the form $ f ( x ) = a ( b ) ^ x $.

So I tried plugging in values so that $ 3 = a ( b ) ^ 2 $ and $ 54 = a ( b ) ^ 5 $ and divided the equations to get that $ b ^ 3 = 18 $. Then since $ b = \sqrt [ 3 ] { 18 } $, I plugged in that value back to $ 3 = a ( b ) ^ 2 $ to find that $ a = \frac { \sqrt { 12 } } { 12 } $. However, when I used my calculators to check the values, the resulting function does not have points $ ( 2 , 3 ) $ and $ ( 5 , 54 ) $. How can I solve this problem?

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Your $b$ is correct, but you should get $a = 3/b^2$ which is not $\sqrt{12}/12$.

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