Please note that lab assignment 2 and onwards **must** be submitted by groups of exactly two people.

In this lab assignment, you will implement the well-known "power" function in
two different new ways. The power function takes two arguments n and k and
computes n^{k}. Your implementation only has to work for non-negative k.

You have possibly seen one implementation of this function in the lecture> If not here it is:

power :: Integer -> Integer -> Integer power n k | k < 0 = error "power: negative argument" power n 0 = 1 power n k = n * power n (k-1)You will implement two more ways in this lab assignment.

*Hint:* "power n 0" takes 1 step.

"power n 1" takes 1 step, and then
uses "power n 0".

"power n 2"
takes 1 step, and then uses "power n 1".

"power n 3"
takes 1 step, and then uses "power n 2".

And so forth.

*Note:* Please make sure you follow the submission guidelines when you write your code.

To calculate "power n k", first construct a list with k elements, all being n, and then use "product".

Implement this idea as a Haskell function "power1".

*Hint:* You have to come up with a way of producing a list with k
elements, all being equal to n. Use a list comprehension, or use the standard
Haskell function "replicate". If you use "replicate", you might want to
use
the function "fromInteger" too! Use Hoogle to find out more about
standard functions (and also to search for standard functions by their type).

We use the fact that, if k is even, we can calculate n^{k} as follows:

nIn other words:^{k}= (n^{2})^{k/2}(k is even)

nSo, instead of recursively using the case for k-1, we use the (much smaller) case for k/2.^{k}= (n * n)^{k/2}(k is even)

If k is not even, we simply go one step down in order to arrive at an even k, just as in the original definition of power:

n^{k}= n * (n^{k-1}) (k is not even)

To sum up, to calculate "power n k":

Implement this idea as a Haskell function "power2".
*Hints:*

**A.** Come up with a number of test cases (inputs you will test your functions on).
Argue why you have chosen these test cases. (Think about for what inputs the
functions are defined, and for what inputs the functions are not defined.)

**B.** Implement two functions: One function "comparePower1" (which given n and k, compares the
result of the "power" function with your "power1" function), and a function
"comparePower2" (which does the same for "power" and "power2").
These two functions will make your testing go easier!

**C.** Write all your test cases as one or two Haskell functions that perform
all test cases. It is probably a good idea to use the functions "comparePower1"
and "comparePower2" here.

*Hint:* You can use a list comprehension to combine all possible cases
you would like to test for n and k. Use the standard Haskell function "and" to
combine the results. If you are not familiar with list
comprehensions, you do not have to use these.

Write your answers in one file, called **Lab1.hs**. For each part, use
Haskell comments to indicate what part of the file contains the answer to that part. For answers in natural language, use English;
write your answers also in Haskell comments. Remove irrelevant things from the
file.

Before you submit your code, Clean It Up! After you feel you are done, spend some time on cleaning your code; make it simpler, remove unneccessary things, etc. We will reject your solution if it is not clean. Clean code:

When you are done, please submit it using the Fire system.

Good luck!