Task: Integer roots
Java: import java.math.BigInteger;

public class IntegerRoots {
    private static BigInteger iRoot(BigInteger base, int n) {
        if (base.compareTo(BigInteger.ZERO) < 0 || n <= 0) {
            throw new IllegalArgumentException();
        }

        int n1 = n - 1;
        BigInteger n2 = BigInteger.valueOf(n);
        BigInteger n3 = BigInteger.valueOf(n1);
        BigInteger c = BigInteger.ONE;
        BigInteger d = n3.add(base).divide(n2);
        BigInteger e = n3.multiply(d).add(base.divide(d.pow(n1))).divide(n2);
        while (!c.equals(d) && !c.equals(e)) {
            c = d;
            d = e;
            e = n3.multiply(e).add(base.divide(e.pow(n1))).divide(n2);
        }
        if (d.compareTo(e) < 0) {
            return d;
        }
        return e;
    }

    public static void main(String[] args) {
        BigInteger b = BigInteger.valueOf(8);
        System.out.print("3rd integer root of 8 = ");
        System.out.println(iRoot(b, 3));

        b = BigInteger.valueOf(9);
        System.out.print("3rd integer root of 9 = ");
        System.out.println(iRoot(b, 3));

        b = BigInteger.valueOf(100).pow(2000).multiply(BigInteger.valueOf(2));
        System.out.print("First 2001 digits of the square root of 2: ");
        System.out.println(iRoot(b, 2));
    }
}

VB: Imports System
Imports System.Numerics
Imports Microsoft.VisualBasic.Strings

Public Module Module1

    Public Function IntSqRoot(v As BigInteger) As BigInteger
        Dim digs As Integer = Math.Max(0, v.ToString().Length / 2 - 1)
        IntSqRoot = BigInteger.Parse("3" & StrDup(digs, "0"))
        Dim term As BigInteger
        Do
            term = v / IntSqRoot
            If Math.Abs(CDbl(term - IntSqRoot)) < 2 Then Exit Do
            IntSqRoot = (IntSqRoot + term) / 2
        Loop Until False
    End Function

    Public Function IntNthRoot(n As Integer, v As BigInteger) As BigInteger
        Dim digs As Integer = Math.Max(0, v.ToString().Length / n - 1)
        IntNthRoot = BigInteger.Parse(If(digs > 1, 3, 2).ToString() & StrDup(digs, "0"))
        Dim va As BigInteger, dr As BigInteger
        Do
            va = v : For i As Integer = 2 To n : va /= IntNthRoot : Next
            va -= IntNthRoot
            dr = va / n : If dr = 0 Then Exit Do
            IntNthRoot += dr
        Loop Until False
    End Function

    Public Sub Main()
        Dim b As BigInteger = BigInteger.Parse("2" & StrDup(4008, "0"))
        Console.WriteLine("Integer Cube Root of 8:")
        Console.WriteLine(IntNthRoot(3, 8).ToString()) 
        Console.WriteLine("Integer Cube Root of 9:")
        Console.WriteLine(IntNthRoot(3, 9).ToString()) 
        Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, square root method):")
        Console.WriteLine(IntSqRoot(b).ToString()) 
        Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, nth root method):")
        Console.WriteLine(IntNthRoot(2, b).ToString()) 
        Console.WriteLine("Integer Cube Root of 2, (actually 2 * 10 ^ 4008):")
        Console.WriteLine(IntNthRoot(3, b).ToString()) 
        b /= 10000
        Console.WriteLine("Integer 7th Root of 2, (actually 2 * 10 ^ 4004):")
        Console.WriteLine(IntNthRoot(7, b).ToString()) 
        If Diagnostics.Debugger.IsAttached Then Console.Read()
    End Sub

End Module