Okay, let me break it down for you in simple terms: this is basically a way of writing really big numbers using arrows and superscripts. But instead of just adding more digits to the end like we normally do (10^3 = 1000), we’re going to use recursion to make things even crazier!

So, let’s say you want to write a number that’s equal to 10 raised to itself 5 times. Instead of writing it out like this:

10^5 = 100000

We can use recursive functional power notation and write it as follows:

10(1)

This might look a little confusing at first, but let me explain what’s going on here. The “” symbol represents the function f that takes an input n (which is 5 in this case), and returns 10 raised to itself n times. So when we write:

f(n) = 10(n)

We’re essentially saying: take the number 10, raise it to itself n times using recursion (i.e., call f with input n-1 and use that result as the base case), and then return the final answer.

Now, let me give you an example of how this works in practice. Let’s say we want to calculate 10 raised to itself 3 times using recursive functional power notation:

f(n) = 10(n)

So when we call f with input n=3, it looks like this:

f(3) = (10(2))(1)

This means that we first calculate the result of calling f with input n=2, which is 10 raised to itself twice using recursion. So when we call f with input n=2, it looks like this:

f(2) = (10(1))(1)

This means that we first calculate the result of calling f with input n=1, which is 10 raised to itself once using recursion. So when we call f with input n=1, it looks like this:

f(1) = (10(0))(1)

This means that we first calculate the result of calling f with input n=0, which is 10 raised to itself zero times using recursion. Since there’s no more recursion left to do at this point, we simply return the base case value:

f(0) = 10(0) = 10^0 = 1

So when we call f with input n=1, it looks like this:

f(1) = (10(0))(1) = 10(0) = 10^3 = 1000

And finally, when we call f with input n=2, it looks like this:

f(2) = (10(1))(1) = (10(0))(1) = 10(0) = 10^6 = 1000000

And when we call f with input n=3, it looks like this:

f(3) = (10(2))(1) = ((10(1))(1))(1) = (10(0))(3) = 10(0) = 10^9 = 100000000

Recursive functional power notation is a powerful tool for writing really big numbers using arrows and superscripts. And while it might look a little intimidating at first, once you get the hang of it, it’s actually pretty easy to use.