What is my exponential potential?
Brachylog, 12 bytes

{^a₀.&amp;b~b^}ᶜ


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Explanation

{^a₀.&amp;b~b^}ᶜ
{         }ᶜ  Count the number of ways the following can succeed:
  a₀            A prefix of
 ^                the first {input} to the power of the second {input}
    .&amp;          produces the same output with the same input as
       ~b         any number
         ^        to the power of
      b           all inputs but the first (i.e. the second input)