Problem 1: compute the exponent of a number.

Input: User inputs the base and exponent, named x and n.

Output: print out the value of xⁿ and how many multiplications which have been executed. How fast can you implement this job (in terms of smallest number of multiplications)?

Test your program on the following five inputs and print out the output for each input.

Input 1: 2 8

Input 2: 0.2 7

Input 3: 6.2 32

Input 4: 2 64

Input 5: 2 63

Problem 2: Insertion Sorting.

We can express insertion sort as a recursive procedure as follows. In order to sort A[1..n], we recursively sort A[1..n-1] and then insert A[n] into the sorted array A[1..n-1]. Write a recursive solution.

Input: 1. User input the number of integers to be sorted, named n.

2. User input n integers.

Output: print out the sorted integers with space between numbers.

Test your program on the following five inputs and print out the output for each input.

Input 1: 1 2 3 4 5 6 7 8

Input 2: 8 7 6 5 4 3 2 1

Input 3: 1 3 5 7 2 4 6 8

Input 4: 5 6 7 8 1 2 3 4

Input 5: 7 8 2 4 1 6 5 3

Note: since the size of the array is one of the inputs, please use dynamic array instead of static array.

Problem 3: Tower of Hanoi.

Write a program to solve the problem of Tower of Hanoi. User would input the number of plates they are going to work on and your program would print the procedure (all the steps of moving the disk from one pole to another pole).

The Tower of Hanoi or Towers of Hanoi (also known as The Towers of Brahma) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks neatly stacked in order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

• Only one disk may be moved at a time.
• Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod.
• No disk may be placed on top of a smaller disk.

The sample output for three disks is:

Enter the number of disks: 3

The sequence of moves involved in the Tower of Hanoi are:

Move disk 1 from rod A to rod C

Move disk 2 from rod A to rod B

Move disk 1 from rod C to rod B

Move disk 3 from rod A to rod C

Move disk 1 from rod B to rod A

Move disk 2 from rod B to rod C

Move disk 1 from rod A to rod C

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