You keep on getting new a new string and continue the loop if it's not empty. Simply insert a control in the loop for an exit string.

while(!s1.equals("exit") && sc.hasNext()) {
    // operate
}

If you want to declare the string inside the loop and not to do the operations in the loop body if the string is "exit":

while(sc.hasNext()) {
    String s1 = sc.next();
    if(s1.equals("exit")) {
        break;
    }
    //operate
}

The nextInt() call does not consume your input (e.g. "abc") if it isn't an integer. So the next time in the loop it still sees the "abc" that you already entered, and that goes on forever. So better use Integer.parseInt(in.next()):

public static void main (String [] args) {

    Scanner in = new Scanner(System.in);
    int k = 0;

    boolean askForInput = true;

    while ( askForInput ) {
        System.out.print("Enter an integer: ");
        try {
            k = Integer.parseInt(in.next());
            askForInput = false;
        } catch (NumberFormatException e) {
            System.out.println("ERR: Not an integer!!");
        }
    }
}

Instead of

while (type == "Y");

use

while (type.equals("Y"));

== checks if two Strings refer to the same object
.equals() checks two objects contain the same data.

Also,it seems that you want to run the loop as long as the user does not enter Y.So change it to:

while (!type.equals("Y"));

you are declaring int question outside the loop and then again inside the loop.

remove the int declaration inside the loop.

In Java the scope of a variable is dependent on which clause it is declare in. If you declare a variable INSIDE a try or a while or many other clauses, that variable is then local to that clause.

Scanner in = new Scanner(System.in);
Scanner choice = new Scanner(System.in);
System.out.println("Do you want to run the program? Y or N");
choice = Scanner.nextLine(); //Error is here
while (choice.equalsIgnoreCase("y"))

The issue is you are improperly using Scanner.

It should be something like:

String userChoice = choice.nextLine();

To use it in your loop more effectively, you can try something like:

String userChoice = choice.nextLine();
while (userChoice.equalsIgnoreCase("y")) {
    // do your loop logic
    // then, to be structured, at the end of your loop, do...
    System.out.print("Do you want to run the program again? Y or N: ");
    userChoice = choice.nextLine();
}

This will re-test the condition after each loop has run. If the user selects "n", the loop will exit naturally.

If you wish to use Scanner, your code can be fixed like that:

public static double sumOfRoots(String numbers, int n){
    Scanner scanner = new Scanner(numbers);
    double sum = 0;
    while(scanner.hasNextDouble())
        sum += Math.pow(scanner.nextDouble(), 1d / n);
    return sum;
}

In Java ^ is not an exponentional operator (as said in comments), so you have to use Math.pow()

One problem is that System.in is basically an infinite stream: hasNext will always return true unless the user enters a special command that closes it.

So you need to have the user enter something that tells you they are done. For example:

while(input.hasNext()) {
    System.out.print("Enter an integer or 'end' to finish: ");
    String next = input.next();
    if("end".equalsIgnoreCase(next)) {
        break;
    }

    int theInt = Integer.parseInt(next);
    ...

For your program, you might have the input you are trying to parse end with a special character like 1,2;3,4;5,6;end or 1,2;3,4;5,6;# that you check for.

And on these lines:

System.out.print(i + ' ');
System.out.print(alist.get(i) + '\n');

It looks like you are trying to perform String concatenation but since char is a numerical type, it performs addition instead. That is why you get the crazy output. So you need to use String instead of char:

System.out.print(i + " ");
System.out.print(alist.get(i) + "\n");

Or just:

System.out.println(i + " " + alist.get(i));

Edit for comment.

You could, for example, pull the input using nextLine from a Scanner with a default delimiter, then create a second Scanner to scan the line:

Scanner sysIn = new Scanner(System.in);
while(sysIn.hasNextLine()) {
    String nextLine = sysIn.nextLine();
    if(nextLine.isEmpty()) {
        break;
    }

    Scanner lineIn = new Scanner(nextLine);
    lineIn.useDelimiter(";|\\,");

    while(lineIn.hasNextInt()) {
        int nextInt = lineIn.nextInt();
        ...
    }
}