Recusive ML function of exponential with three variable helper function

Question

I need help figuring out what I need to do for the helper function recursively, I am kinda lost of in what I need to do for the helper function.

Here is the question and the example input.

Here is the example of what the helper function does

This is what I have written so far

Answer 1

You only need to change the notation slightly to find the solution.

Instead of a*bⁿ, write exp3(a, b, n).

Then the example says,

  exp(3,4)
= exp3(1, 3, 4)
= exp3(1*3, 3, 4-1)
= exp3(3*3, 3, 3-1)
= exp3(3*9, 3, 2-1)
= exp3(3*27, 3, 1-1)
= 81

And you have actually been given the solution – the text literally says "Lemma 4.15 provides a base case, Lemma 4.16 a recursive case".

Lemma 4.15:

exp3(a, b, 0) = a

Lemma 4.16:

exp3(a, b, n) = exp3(a * b, b, n - 1)

And the function that needs help should only have one case; the first "step" shown in the example:

fun exp(a, b) = exp3(1, a, b)