# HOFs - Procedures as Returned Values

We've seen how to write a procedure that takes another procedure as an argument. It turns out we can do the opposite as well - we can create a procedure that returns a procedure! Returning procedures is a great way to abstract even further. Instead of creating the procedure directly, we can have a program that creates the procedure for us! Depending on what arguments we give the program, it can create many different procedures.

## Example: make-power

(We're not actually making power. That'd be powerplaying ;).)

Let's say we want to define a procedure sum-powers that takes the nth power of everything number between a and b and sums them together. We already have our procedure, reproduced below:

(define (sum f a b)
(if (> a b)
0
(+ (f a) (sum f (+ a 1) b))))


From what we learned so far, it'd look something like this:

(define (sum-powers n a b)
(sum (lambda (x) (expt x n)) a b))


But what if we create a new function called make-power, that, given a power n, returns a function that takes a number x and returns its nth power? It looks like this:

(define (make-power n)
(lambda (x) (expt x n)))


As we noted earlier, lambdas return functions. This means that if we define the call to make-power as a lambda, it will return a function! We can now do this:

(define square (make-power 2))
(define cube (make-power 3))


And we can rewrite our sum-powers function like this:

(define (sum-powers n a b)
(sum (make-power n) a b))


Note also how much we've progressed in abstraction. At the beginning of this lab, we defined a different procedure for each different type of sum: sum-doubles, sum-squares, and sum-cubes.

But now, we have abstracted the summation itself, so that we can express any summation in a single clear line.