Integer roots

Integer roots is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Create a program that computes an approximation of the principal   N^th   root of   X   as the largest integer less than or equal to   R   for which   R^N=X.

──where:

       N  is a positive integer. 
       X  is a non-negative integer. 
       R  (the root)   is a non-negative real number. 

No arbitrary limits should be placed on the magnitudes of the numbers involved.

Example:   With   N=3   and   X=8   you would calculate the number   2   because   $2^{3}=8$

Example:   With   N=3   and   X=9  you would again calculate the number   2   because 2 is the largest integer less than or equal to the root   R.

Example:   With   N=2   and   X=2×100^2,000   you would calculate a large integer consisting of the first   2,001   digits (in order) of the square root of two.

11l

Translation of: D

F iRoot(BigInt b, Int n)
   I b < 2 {R b}
   V n1 = n - 1
   V n2 = BigInt(n)
   V n3 = BigInt(n1)
   V c = BigInt(1)
   V d = (n3 + b) I/ n2
   V e = (n3 * d + b I/ d ^ n1) I/ n2
   L c != d & c != e
      c = d
      d = e
      e = (n3 * e + b I/ e ^ n1) I/ n2
   I d < e {R d}
   R e

print(‘3rd root of 8 = ’iRoot(8, 3))
print(‘3rd root of 9 = ’iRoot(9, 3))
print(‘First 2001 digits of the square root of 2: ’iRoot(BigInt(100) ^ 2000 * 2, 2))

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Arturo

Translation of: D

iroot: function [b n][
    if b<2 -> return b
 
    n1: n-1
    n2: n
    n3: n1
    c: 1
    d: (n3+b)/n2
    e: ((n3*d) + b/d^n1)/n2
    while [and? c<>d c<>e][
        c: d
        d: e
        e: ((n3*e) + b/e^n1)/n2
    ]
    if d<e -> return d
    return e
]
 
print ["3rd root of 8:" iroot 8 3]
print ["3rd root of 9:" iroot 9 3]
print ["First 2001 digits of the square root of 2:" iroot (100^2000)*2 2]

Output:

3rd root of 8: 2 
3rd root of 9: 2 
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

BASIC256

Translation of: FreeBASIC

function root(n, x)
	for nr = floor(sqr(x)) to 1 step -1
		if (nr ^ n) <= x then return nr
	next nr
end function

print root(3, 8)
print root(3, 9)
print root(4, 167)
print root(2, 2e18)
end

Output:

Igual que la entrada de FreeBASIC.

C

Translation of: C++

#include <stdio.h>
#include <math.h>

typedef unsigned long long ulong;

ulong root(ulong base, ulong n) {
    ulong n1, n2, n3, c, d, e;

    if (base < 2) return base;
    if (n == 0) return 1;

    n1 = n - 1;
    n2 = n;
    n3 = n1;
    c = 1;
    d = (n3 + base) / n2;
    e = (n3 * d + base / (ulong)powl(d, n1)) / n2;

    while (c != d && c != e) {
        c = d;
        d = e;
        e = (n3*e + base / (ulong)powl(e, n1)) / n2;
    }

    if (d < e) return d;
    return e;
}

int main() {
    ulong b = (ulong)2e18;

    printf("3rd root of 8 = %lld\n", root(8, 3));
    printf("3rd root of 9 = %lld\n", root(9, 3));
    printf("2nd root of %lld = %lld\n", b, root(b, 2));

    return 0;
}

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2000000000000000000 = 1414213562

C#

Translation of: Java

using System;
using System.Numerics;

namespace IntegerRoots {
    class Program {
        static BigInteger IRoot(BigInteger @base, int n) {
            if (@base < 0 || n <= 0) {
                throw new ArgumentException();
            }

            int n1 = n - 1;
            BigInteger n2 = n;
            BigInteger n3 = n1;
            BigInteger c = 1;
            BigInteger d = (n3 + @base) / n2;
            BigInteger e = ((n3 * d) + (@base / BigInteger.Pow(d, n1))) / n2;
            while (c != d && c != e) {
                c = d;
                d = e;
                e = (n3 * e + @base / BigInteger.Pow(e, n1)) / n2;
            }
            if (d < e) {
                return d;
            }
            return e;
        }

        static void Main(string[] args) {
            Console.WriteLine("3rd integer root of 8 = {0}", IRoot(8, 3));
            Console.WriteLine("3rd integer root of 9 = {0}", IRoot(9, 3));

            BigInteger b = BigInteger.Pow(100, 2000) * 2;
            Console.WriteLine("First 2001 digits of the sqaure root of 2: {0}", IRoot(b, 2));
        }
    }
}

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the sqaure root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

C++

#include <iostream>
#include <math.h>

unsigned long long root(unsigned long long base, unsigned int n) {
	if (base < 2) return base;
	if (n == 0) return 1;

	unsigned int n1 = n - 1;
	unsigned long long n2 = n;
	unsigned long long n3 = n1;
	unsigned long long c = 1;
	auto d = (n3 + base) / n2;
	auto e = (n3 * d + base / pow(d, n1)) / n2;

	while (c != d && c != e) {
		c = d;
		d = e;
		e = (n3*e + base / pow(e, n1)) / n2;
	}

	if (d < e) return d;
	return e;
}

int main() {
	using namespace std;

	cout << "3rd root of 8 = " << root(8, 3) << endl;
	cout << "3rd root of 9 = " << root(9, 3) << endl;

	unsigned long long b = 2e18;
	cout << "2nd root of " << b << " = " << root(b, 2) << endl;

	return 0;
}

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2000000000000000000 = 1414213562

D

Translation of: Kotlin

import std.bigint;
import std.stdio;

auto iRoot(BigInt b, int n) in {
    assert(b >=0 && n > 0);
} body {
    if (b < 2) return b;
    auto n1 = n - 1;
    auto n2 = BigInt(n);
    auto n3 = BigInt(n1);
    auto c = BigInt(1);
    auto d = (n3 + b) / n2;
    auto e = (n3 * d + b / d^^n1) / n2;
    while (c != d && c != e) {
        c = d;
        d = e;
        e = (n3 * e + b / e^^n1) / n2;
    }
    if (d < e) return d;
    return e;
}

void main() {
    auto b = BigInt(8);
    writeln("3rd root of 8 = ", b.iRoot(3));
    b = BigInt(9);
    writeln("3rd root of 9 = ", b.iRoot(3));
    b = BigInt(100)^^2000*2;
    writeln("First 2001 digits of the square root of 2: ", b.iRoot(2));
}

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Elixir

Translation of: Ruby

defmodule Integer_roots do
  def root(_, b) when b<2, do: b
  def root(a, b) do
    a1 = a - 1
    f = fn x -> (a1 * x + div(b, power(x, a1))) |> div(a) end
    c = 1
    d = f.(c)
    e = f.(d)
    until(c, d, e, f)
  end
  
  defp until(c, d, e, _) when c in [d, e], do: min(d, e)
  defp until(_, d, e, f), do: until(d, e, f.(e), f)
  
  defp power(_, 0), do: 1
  defp power(n, m), do: Enum.reduce(1..m, 1, fn _,acc -> acc*n end)
  
  def task do
    IO.puts root(3,8)
    IO.puts root(3,9)
    IO.puts "First 2,001 digits of the square root of two:"
    IO.puts root(2, 2 * power(100, 2000))
  end
end

Integer_roots.task

Output:

2
2
First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

F#

Translation of: C#

open System

let iroot (base_ : bigint) n =
    if base_ < bigint.Zero || n <= 0 then
        raise (ArgumentException "Bad parameter")

    let n1 = n - 1
    let n2 = bigint n
    let n3 = bigint n1
    let mutable c = bigint.One
    let mutable d = (n3 + base_) / n2
    let mutable e = ((n3 * d) + (base_ / bigint.Pow(d, n1))) / n2
    while c <> d && c <> e do
        c <- d
        d <- e
        e <- (n3 * e + base_ / bigint.Pow(e, n1)) / n2

    if d < e then
        d
    else
        e

[<EntryPoint>]
let main _ =
    Console.WriteLine("3rd integer root of 8 = {0}", (iroot (bigint 8) 3))
    Console.WriteLine("3rd integer root of 9 = {0}", (iroot (bigint 9) 3))

    let b = bigint.Pow(bigint 100, 2000) * (bigint 2)
    Console.WriteLine("First 2001 digits of the sqaure root of 2: {0}", (iroot b 2))

    0 // return an integer exit code

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the sqaure root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Factor

Translation of: Sidef

USING: io kernel locals math math.functions math.order
prettyprint sequences ;

:: (root) ( a b -- n )
    a 1 - 1 :> ( a1 c! )
    [| x | a1 x * b x a1 ^ /i + a /i ] :> f
    c f call :> d!
    d f call :> e!
    [ c { d e } member? ] [
        d c! e d! e f call e!
    ] until
    d e min ;

: root ( a b -- n ) dup 2 < [ nip ] [ (root) ] if ;

"First 2,001 digits of the square root of two:" print
2 100 2000 ^ 2 * root .

Output:

First 2,001 digits of the square root of two:
14142135623730950488016887242096980[...]32952546758516447107578486024636008

FreeBASIC

Translation of: Ring

#define floor(x) ((x*2.0-0.5) Shr 1)

Function root(n As Uinteger, x As Uinteger) As Uinteger
    For nr As Uinteger = floor(Sqr(x)) To 1 Step -1
        If (nr ^ n) <= x Then Return nr
    Next nr
End Function

Print root(3, 8) 
Print root(3, 9) 
Print root(4, 167)
Print root(2, 2e18)

Sleep

Output:

2
2
3
1414213562

Go

int

package main

import "fmt"

func main() {
    fmt.Println(root(3, 8))
    fmt.Println(root(3, 9))
    fmt.Println(root(2, 2e18))
}

func root(N, X int) int {
    // adapted from https://en.wikipedia.org/wiki/Nth_root_algorithm
    for r := 1; ; {
        x := X
        for i := 1; i < N; i++ {
            x /= r
        }
        x -= r
        // A small complication here is that Go performs truncated integer
        // division but for negative values of x, Δr in the line below needs 
        // to be computed as the floor of x / N.  The following % test and
        // correction completes the floor division operation (for positive N.)
        Δr := x / N
        if x%N < 0 {
            Δr--
        }
        if Δr == 0 {
            return r
        }
        r += Δr
    }
}

Output:

2
2
1414213562

big.Int

package main

import (
    "fmt"
    "math/big"
)

func main() {
    fmt.Println(root(3, "8"))
    fmt.Println(root(3, "9"))
    fmt.Println(root(2, "2000000000000000000"))
    fmt.Println(root(2, "200000000000000000000000000000000000000000000000000"))
}

var one = big.NewInt(1)

func root(N int, X string) *big.Int {
    var xx, x, Δr big.Int
    xx.SetString(X, 10)
    nn := big.NewInt(int64(N))
    for r := big.NewInt(1); ; {
        x.Set(&xx)
        for i := 1; i < N; i++ {
            x.Quo(&x, r)
        }
        // big.Quo performs Go-like truncated division and would allow direct
        // translation of the int-based solution, but package big also provides
        // Div which performs Euclidean rather than truncated division.
        // This gives the desired result for negative x so the int-based
        // correction is no longer needed and the code here can more directly
        // follow the Wikipedia article.
        Δr.Div(x.Sub(&x, r), nn)
        if len(Δr.Bits()) == 0 {
            return r
        }
        r.Add(r, &Δr)
    }
}

Output:

2
2
1414213562
14142135623730950488016887

Haskell

Translation of: Python

root :: Integer -> Integer -> Integer
root a b = findAns $ iterate (\x -> (a1 * x + b `div` (x ^ a1)) `div` a) 1
  where
    a1 = a - 1
    findAns (x:xs@(y:z:_))
      | x == y || x == z = min y z
      | otherwise = findAns xs

main :: IO ()
main = do
  print $ root 3 8
  print $ root 3 9
  print $ root 2 (2 * 100 ^ 2000) -- first 2001 digits of the square root of 2

Or equivalently, in terms of an applicative expression:

integerRoot :: Integer -> Integer -> Integer
integerRoot n x =
  go $ iterate ((`div` n) . ((+) . (pn *) <*> (x `div`) . (^ pn))) 1
  where
    pn = pred n
    go (x:xs@(y:z:_))
      | x == y || x == z = min y z
      | otherwise = go xs
 
main :: IO ()
main = mapM_ (print . uncurry integerRoot) [(3, 8), (3, 9), (2, 2 * 100 ^ 2000)]

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

J

<.@%: satisfies this task. Left argument is the task's N, right argument is the task's X:

Note: Depending on N, one must select the proper number of digits, that is, 2000, 2001, 2002, etc..., otherwise the result will be the digits of the nth root of 20, 2000, etc...
For example, If you use "3 <.@%: (2*10x^2*2000)" instead of "3 <.@%: (2*10x^2*2001)", you will get an output starting with "271441761659490657151808946...", which are the first digits of the cube root of 20, not 2.

   9!:37]0 4096 0 222 NB. set display truncation sufficiently high for our results

   2 <.@%: (2*10x^2*2000)
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008
   3 <.@%: (2*10x^2*2001)
125992104989487316476721060727822835057025146470150798008197511215529967651395948372939656243625509415431025603561566525939902404061373722845911030426935524696064261662500097747452656548030686718540551868924587251676419937370969509838278316139915512931369536618394746344857657030311909589598474110598116290705359081647801147352132548477129788024220858205325797252666220266900566560819947156281764050606648267735726704194862076214429656942050793191724414809204482328401274703219642820812019057141889964599983175038018886895942020559220211547299738488026073636974178877921579846750995396300782609596242034832386601398573634339097371265279959919699683779131681681544288502796515292781076797140020406056748039385612517183570069079849963419762914740448345402697154762285131780206438780476493225790528984670858052862581300054293885607206097472230406313572349364584065759169169167270601244028967000010690810353138529027004150842323362398893864967821941498380270729571768128790014457462271477023483571519055067220848184850092872392092826466067171742477537097370300127429180940544256965920750363575703751896037074739934610144901451576359604711119738452991329657262589048609788561801386773836157730098659836608059757560127871214868562426845564116515581793532280158962912994450040120842541416015752584162988142309735821530604057724253836453253356
   5 <.@%: (2*10x^2*2000)
114869835499703500679862694677792758944385088909779750551371111849360320625351305681147311301150847391457571782825280872990018972855371267615994917020637676959403854539263226492033301322122190625130645468320078386350285806907949085127708283982797043969640382563667945344431106523789654147255972578315704103326302050272017414235255993151553782375173884359786924137881735354092890268530342009402133755822717151679559278360263800840317501093689917495888199116488588871447782240220513546797235647742625493141141704109917646404017146978939243424915943739448283626010758721504375406023613552985026793701507511351368254645700768390780390334017990233124030682358360249760098999315658413563173197024899154512108923313999675829872581317721346549115423634135836394159076400636688679216398175376716152621781331348
   7 <.@%: (2*10x^2*2002)
1104089513673812337649505387623344721325326600780124165514532464142106322880380980716598289886302005146897159065579931253969214680430855796510648058388081961639198643922155838145512343974763395078906646859029211806139421440562835192195007740110439139292223389537903767320705032063903809884944457070845279252405827307254864679671836816589429995916822424590361601902611505690284386526869351720866524568004847701822070064334667580822044823960984514550922242408608825451442062850448298384317793721518676765230683406727811327252052334859250776811047221310365241746671294399050316

Java

Translation of: Kotlin

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));
    }
}

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

jq

Adapted from Julia

Works with gojq, the Go implementation of jq

# To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);

# If $j is 0, then an error condition is raised;
# otherwise, assuming infinite-precision integer arithmetic,
# if the input and $j are integers, then the result will be an integer.
def idivide($j):
  (. - (. % $j)) / $j ;

def iroot(a; b):
  if b < 2 then b
  else (a-1) as $a1
  | {c: 1,
     d: (($a1 + (b | idivide(1))) | idivide(a)) }
  | .d as $d
  | .e = ($a1 * $d + (b |idivide($d | power($a1))) | idivide(a))
  | until( .d == .c or .c == .e; 
       .c = .d
       | .d = .e
       | .e as $e
       | .e = ($a1 * .e + (b | idivide(($e | power($a1)))) | idivide(a)) )
  | [.d, .e] | min
  end ;

# The task: 
"First 2,001 digits of the square root of two:", 
iroot(2; 2 * (100 | power(2000)))

Output:

Exactly as for Julia.

Julia

Works with: Julia version 1.3

Translation of: Python

function iroot(a, b)
    b < 2 && return b
    a1, c = a - 1, 1
    d = (a1 * c + b ÷ c^a1) ÷ a
    e = (a1 * d + b ÷ d^a1) ÷ a
    while d ≠ c ≠ e
        c, d, e = d, e, (a1 * e + b ÷ (e ^ a1)) ÷ a
    end
 
    min(d, e)
end

println("First 2,001 digits of the square root of two:\n", iroot(2, 2 * big(100) ^ 2000))

Output:

First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Kotlin

Translation of: Python

// version 1.1.2

import java.math.BigInteger

val bigZero = BigInteger.ZERO
val bigOne  = BigInteger.ONE
val bigTwo  = BigInteger.valueOf(2L)

fun BigInteger.iRoot(n: Int): BigInteger {
    require(this >= bigZero && n > 0)
    if (this < bigTwo) return this
    val n1 = n - 1
    val n2 = BigInteger.valueOf(n.toLong())
    val n3 = BigInteger.valueOf(n1.toLong())
    var c = bigOne
    var d = (n3 + this) / n2
    var e = (n3 * d + this / d.pow(n1)) / n2
    while (c != d && c != e) {
        c = d
        d = e
        e = (n3 * e + this / e.pow(n1)) / n2
    }
    return if (d < e) d else e
}    

fun main(args: Array<String>) {
    var b: BigInteger
    b = BigInteger.valueOf(8L)
    println("3rd integer root of 8 = ${b.iRoot(3)}\n")
    b = BigInteger.valueOf(9L)
    println("3rd integer root of 9 = ${b.iRoot(3)}\n")    
    b = BigInteger.valueOf(100L).pow(2000) * bigTwo
    println("First 2001 digits of the square root of 2:")
    println(b.iRoot(2))
}

Output:

3rd integer root of 8 = 2

3rd integer root of 9 = 2

First 2001 digits of the square root of 2:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Lua

Translation of: C

function root(base, n)
    if base < 2 then return base end
    if n == 0 then return 1 end

    local n1 = n - 1
    local n2 = n
    local n3 = n1
    local c = 1
    local d = math.floor((n3 + base) / n2)
    local e = math.floor((n3 * d + base / math.pow(d, n1)) / n2)

    while c ~= d and c ~= e do
        c = d
        d = e
        e = math.floor((n3 * e + base / math.pow(e, n1)) / n2)
    end

    if d < e then return d end
    return e
end

-- main
local b = 2e18

print("3rd root of 8 = " .. root(8, 3))
print("3rd root of 9 = " .. root(9, 3))
print("2nd root of " .. b .. " = " .. root(b, 2))

Output:

3rd root of 8 = 2
3rd root of 9 = 2
2nd root of 2e+018 = 1414213562

Modula-2

MODULE IntegerRoot;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,ReadChar;

PROCEDURE pow(b : LONGCARD; p : CARDINAL) : LONGCARD;
VAR
    result : LONGCARD;
BEGIN
    result := 1;
    WHILE p > 0 DO
        IF p MOD 2 = 1 THEN
            DEC(p);
            result := result * b;
        END;
        p := p / 2;
        b := b * b
    END;
    RETURN result
END pow;

PROCEDURE root(base : LONGCARD; n : CARDINAL) : LONGCARD;
VAR
    n1,n2,n3,c,d,e : LONGCARD;
BEGIN
    IF base < 2 THEN RETURN base END;
    IF n = 0 THEN RETURN 1 END;

    n1 := n - 1;
    n2 := n;
    n3 := n1;
    c := 1;
    d := (n3 + base) / n2;
    e := (n3 * d + base / pow(d, n1)) / n2;

    WHILE (c # d) AND (c # e) DO
        c := d;
        d := e;
        e := (n3 * e + base / pow(e, n1)) / n2
    END;

    IF d < e THEN RETURN d END;
    RETURN e
END root;

(* main *)
VAR
    buf : ARRAY[0..63] OF CHAR;
    b : LONGCARD;
BEGIN
    FormatString("3rd root of 8 = %u\n", buf, root(8, 3));
    WriteString(buf);

    FormatString("3rd root of 9 = %u\n", buf, root(9, 3));
    WriteString(buf);

    b := 2000000000000000000;
    FormatString("2nd root of %u = %u\n", buf, b, root(b, 2));
    WriteString(buf);

    ReadChar
END IntegerRoot.

Nim

Translation of: Kotlin

Library: bignum

import bignum

proc root(x: Int; n: int): Int =
  if x < 2: return x
  let n1 = (n - 1).culong
  var c = newInt(1)
  var d = (n1 + x) div n
  var e = (n1 * d + x div d.pow(n1)) div n
  while c != d and c != e:
    c = d
    d = e
    e = (n1 * e + x div e.pow(n1)) div n
  result = if d < e: d else: e


var x: Int
x = newInt(8)
echo "3rd integer root of 8 = ", x.root(3)
x = newInt(9)
echo "3rd integer root of 9 = ", x.root(3)
x = newInt(100).pow(2000) * newInt(2)
echo "First 2001 digits of the square root of 2:"
let s = $x.root(2)
for i in countup(0, s.high, 87): echo s.substr(i, i + 86)

Output:

3rd integer root of 8 = 2
3rd integer root of 9 = 2
First 2001 digits of the square root of 2:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038
753432764157273501384623091229702492483605585073721264412149709993583141322266592750559
275579995050115278206057147010955997160597027453459686201472851741864088919860955232923
048430871432145083976260362799525140798968725339654633180882964062061525835239505474575
028775996172983557522033753185701135437460340849884716038689997069900481503054402779031
645424782306849293691862158057846311159666871301301561856898723723528850926486124949771
542183342042856860601468247207714358548741556570696776537202264854470158588016207584749
226572260020855844665214583988939443709265918003113882464681570826301005948587040031864
803421948972782906410450726368813137398552561173220402450912277002269411275736272804957
381089675040183698683684507257993647290607629969413804756548237289971803268024744206292
691248590521810044598421505911202494413417285314781058036033710773091828693147101711116
839165817268894197587165821521282295184884720896946338628915628827659526351405422676532
396946175112916024087155101351504553812875600526314680171274026539694702403005174953188
629256313851881634780015693691768818523786840522878376293892143006558695686859645951555
016447245098368960368873231143894155766510408839142923381132060524336294853170499157717
562285497414389991880217624309652065642118273167262575395947172559346372386322614827426
222086711558395999265211762526989175409881593486400834570851814722318142040704265090565
323333984364578657967965192672923998753666172159825788602633636178274959942194037777536
814262177387991945513972312740668983299898953867288228563786977496625199665835257761989
393228453447356947949629521688914854925389047558288345260965240965428893945386466257449
275563819644103169798330618520193793849400571563337205480685405758679996701213722394758
214263065851322174088323829472876173936474678374319600015921888073478576172522118674904
249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

PARI/GP

sqrtnint(8,3)
sqrtnint(9,3)
sqrtnint(2*100^2000,2)

Output:

%1 = 2
%2 = 2
%3 = 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Perl

Translation of: Ruby

use bigint;

sub integer_root {
    our($a,$b) = @_;
    our $a1 = $a - 1;
    my $c = 1;
    my $d = f($c);
    my $e = f($d);
    ($c, $d, $e) = ($d, $e, f($e)) until $c==$d || $c==$e;
    return $d < $e ? $d : $e;

    sub f { ($a1*$_[0]+$b/$_[0]**$a1)/$a }
}

print integer_root( 3, 8), "\n";
print integer_root( 3, 9), "\n";
print integer_root( 2, 2 * 100 ** 2000), "\n";

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Using a module

If using bigints, we can do this directly, which will be much faster than the method above:

use bigint;
print 8->babs->broot(3),"\n";
print 9->babs->broot(3),"\n";
print +(2*100**2000)->babs->broot(2),"\n";

The babs calls are only necessary if the input might be non-negative.

Even faster, using a module:

use bigint;
use ntheory "rootint";
print rootint(8,3),"\n";
print rootint(9,3),"\n";
print rootint(2*100**2000,2),"\n";

Both generate the same output as above.

Phix

Library: Phix/mpfr

with javascript_semantics 
include mpfr.e

function integer_root(integer n, object a)
-- a must be integer or string
--  (were a an mpz you would have to invoke mpz_init_set(), not mpz_init(),
--   or better yet pass a as the second parameter of mpz_root() instead.)
    mpz res = mpz_init(a)
    mpz_nthroot(res,res,n)
    return mpz_get_str(res)
end function
 
atom t0 = time()
printf(1,"3rd root of 8 = %s\n", {integer_root(3,8)})
printf(1,"3rd root of 9 = %s\n", {integer_root(3,9)})
string s = integer_root(2,"2"&repeat('0',4000))
printf(1,"First digits of the square root of 2: %s\n", {shorten(s)})
s = integer_root(3,"2"&repeat('0',6000))
printf(1,"First digits of the  cube  root of 2: %s\n", {shorten(s)})
?elapsed(time()-t0)

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First digits of the square root of 2: 14142135623730950488...47107578486024636008 (2,001 digits)
First digits of the  cube  root of 2: 12599210498948731647...22546828353183047061 (2,001 digits)
"0.4s"

While this finishes near-instantly on the desktop, it takes about 25s under pwa/p2js.

Python

def root(a, b):
    if b < 2:
        return b
    a1 = a - 1
    c = 1
    d = (a1 * c + b // (c ** a1)) // a
    e = (a1 * d + b // (d ** a1)) // a
    while c not in (d, e):
        c, d, e = d, e, (a1 * e + b // (e ** a1)) // a
    return min(d, e)


print("First 2,001 digits of the square root of two:\n{}".format(
    root(2, 2 * 100 ** 2000)
))

Output:

First 2,001 digits of the square root of two:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Quackery

Translation of: Python

  [ stack ]                    is a-1        (     --> s )
  [ stack ]                    is b          (     --> s )

  [ a-1 share tuck 2dup *
    unrot **
    b share swap / +
    swap 1+ / ]                is nextapprox (   n --> n )
 
  [ over 2 < iff drop done
    1 - a-1 put
    b put
    1
    2 times [ dup nextapprox ]
    [ dip [ 2dup = rot ]
      tuck = rot or not while
      dup nextapprox again ]
    min
    b release a-1 release ]    is root       ( n n --> n )

  say "3rd root of 8 = " 8 3 root echo cr
  say "3rd root of 9 = " 9 3 root echo cr
  say "First 2001 digits of the square root of 2: "
  2 100 2000 ** * 2 root echo cr

Output:

3rd root of 8 = 2
3rd root of 9 = 2
First 2001 digits of the square root of 2: 141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Racket

See #Scheme, there’s very little can be done to improve it.

Raku

(formerly Perl 6)

Translation of: Python

sub integer_root ( Int $p where * >= 2, Int $n --> Int ) {
    my Int $d = $p - 1;
    my $guess = 10**($n.chars div $p);
    my $iterator = { ( $d * $^x   +   $n div ($^x ** $d) ) div $p };
    my $endpoint = {      $^x      ** $p <= $n
                     and ($^x + 1) ** $p >  $n };
    min (+$guess, $iterator ... $endpoint)[*-1, *-2];
}

say integer_root( 2, 2 * 100 ** 2000 );

Output:

141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

REXX

No error checking is performed to ensure the root is a non-zero integer.

This version incorporates some optimization when computing square roots   (because   M   is unity,   there is no need to
multiply the guess [G] by unity,   and no need to compute the guess to the 1^st power,   bypassing some trivial arithmetic).

integer result only

/*REXX program calculates the Nth root of a number to a specified number of decimal digs*/
parse arg num root digs .                        /*obtain the optional arguments from CL*/
if  num=='' |  num==","  then  num=   2          /*Not specified?  Then use the default.*/
if root=='' | root==","  then root=   2          /* "      "         "   "   "     "    */
if digs=='' | digs==","  then digs=2001          /* "      "         "   "   "     "    */
numeric digits digs                              /*utilize this number of decimal digits*/
say 'number='  num                               /*display the number that will be used.*/
say '  root='  root                              /*   "     "    root   "    "   "   "  */
say 'digits='  digs                              /*   "    dec. digits  "    "   "   "  */
say                                              /*   "    a blank line.                */
say 'result:';       say rootI(num, root, digs)  /*   "    what it is; display the root.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
rootI: procedure; parse arg x,root,p             /*obtain the numbers,  Y is the root #.*/
       numeric digits p*root+length(x)           /*double the number of digits  + guard.*/
       if x<2  then return x                     /*B is one or zero?  Return that value.*/
       z=x*(10**root)**p                         /*calculate the number with appended 0s*/
       m=root - 1                                /*utilize a diminished (by one) power. */
       g=(1 + z) % root                          /*take a stab at the first root guess. */
       old=.                                     /* [↓]  When M=1, a fast path for sqrt.*/
       if m==1  then  do  until old==g;   old=g;     g=(g   + z %  g     )  % root;    end
                else  do  until old==g;   old=g;     g=(g*m + z % (g**m) )  % root;    end
       return left(g,p)                          /*return the  Nth root of Z to invoker.*/

output   when the defaults are being used:

number= 2
  root= 2
digits= 2001

result:
14142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714
70109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847
16038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016
20758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342194897278290641045072636881313739855256117322040245091227700226941127573627280495738108967504018369
86836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847
20896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558
69568685964595155501644724509836896036887323114389415576651040883914292338113206052433629485317049915771756228549741438999188021762430965206564211827316726257539594717255934637238632261482742622208671
15583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689
83299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685
40575867999670121372239475821426306585132217408832382947287617393647467837431960001592188807347857617252211867490424977366929207311096369721608933708661156734585334833295254675851644710757848602463600
8

true results

Negative and complex roots are supported.   The expressed root may have a decimal point.

/*REXX program calculates the Nth root of a number to a specified number of decimal digs*/
parse arg num root digs .                        /*obtain the optional arguments from CL*/
if  num=='' |  num==","  then  num=   2          /*Not specified?  Then use the default.*/
if root=='' | root==","  then root=   2          /* "      "         "   "   "     "    */
if digs=='' | digs==","  then digs=2001          /* "      "         "   "   "     "    */
numeric digits digs                              /*utilize this number of decimal digits*/
say 'number='  num                               /*display the number that will be used.*/
say '  root='  root                              /*   "     "    root   "    "   "   "  */
say 'digits='  digs                              /*   "    dec. digits  "    "   "   "  */
say                                              /*   "    a blank line.                */
say 'result:';           say iRoot(num, root)    /*   "    what it is; display the root.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
iRoot: procedure; parse arg x 1 ox,    y 1 oy    /*obtain the numbers,  Y is the root #.*/
i=;                         x=abs(x);  y=abs(y)  /*use the absolute values of  X and Y. */
if ox<0 & oy//2==0  then do;  i='i';  ox=x;  end /*if the results will be imaginary ··· */
od=digits()                                      /*the current number of decimal digits.*/
a=od+9                                           /*bump the decimal digits by  nine.    */
numeric form                                     /*number will be in  exponential  form.*/
parse value format(x,2,1,,0) 'E0' with ? 'E' _ . /*obtain exponent so we can do division*/
g=(?/y'E'_ % y)  +  (x>1)                        /*this is a best first guess of a root.*/
m=y-1                                            /*define a (fast) variable for later.  */
d=5                                              /*start with only five decimal digits. */
             do  until d==a                      /*keep computing 'til we're at max digs*/
             d=min(d+d,a);           dm=d-2      /*bump number of (growing) decimal digs*/
             numeric digits d                    /*increase the number of decimal digits*/
             o=0                                 /*set the old value to zero (1st time).*/
                 do  until o=g;      o=g         /*keep computing as long as  G changes.*/
                 g=format((m*g**y+x)/y/g**m,,dm) /*compute the  Yth  root of  X.        */
                 end   /*until o=g*/
             end       /*until d==a*/
_=g*sign(ox)                                     /*change the sign of the result, maybe.*/
numeric digits od                                /*set  numeric digits  to the original.*/
if oy<0  then return (1/_)i                      /*Is the root negative?  Use reciprocal*/
              return (_/1)i                      /*return the  Yth root of X to invoker.*/

output   when the defaults are being used:

number= 2
  root= 2
digits= 2001

result:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571
47010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884
71603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372352885092648612494977154218334204285686060146824720771435854874155657069677653720226485447015858801
62075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836
98683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884
72089694633862891562882765952635140542267653239694617511291602408715510135150455381287560052631468017127402653969470240300517495318862925631385188163478001569369176881852378684052287837629389214300655
86956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867
11558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668
98329989895386728822856378697749662519966583525776198939322845344735694794962952168891485492538904755828834526096524096542889394538646625744927556381964410316979833061852019379384940057156333720548068
54057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360
08

output   when using the input of:   -81

number= -81
  root= 2
digits= 2001

result:
9i

output   when using the input of:   4   -2

number= 4
  root= -2
digits= 2001

result:
0.5

Ring

# Project : Integer roots

see root(3, 8) 
see root(3, 9) 
see root(4, 167)

func root(n, x)
       for nr = floor(sqrt(x)) to 1 step -1
            if pow(nr, n) <= x
               see nr + nl
               exit
            ok
       next

Output:

2
2
3

Ruby

Translation of: Python, zkl

def root(a,b)
  return b if b<2
  a1, c = a-1, 1
  f = -> x {(a1*x+b/(x**a1))/a}  # a lambda with argument x
  d = f[c]
  e = f[d]
  c, d, e = d, e, f[e] until [d,e].include?(c)
  [d,e].min
end

puts "First 2,001 digits of the square root of two:"
puts root(2, 2*100**2000)

Output:

First 2,001 digits of the square root of two:
14142135623730950488016887242096(...)46758516447107578486024636008

Rust

The rug crate provides the functionality required for this task.

// [dependencies]
// rug = "1.9"

fn shorten(s: &str, digits: usize) -> String {
    if s.len() <= digits + 3 {
        return String::from(s);
    }    
    format!("{}...{}", &s[0..digits/2], &s[s.len()-digits/2..])
}

fn main() {
    use rug::{ops::Pow, Integer};

    let x = Integer::from(8);
    let r = Integer::from(x.root_ref(3));
    println!("Integer cube root of {}: {}", x, r);

    let x = Integer::from(9);
    let r = Integer::from(x.root_ref(3));
    println!("Integer cube root of {}: {}", x, r);

    let mut x = Integer::from(100).pow(2000);
    x *= 2;
    let s = Integer::from(x.root(2)).to_string();
    println!("First {} digits of the square root of 2:\n{}", s.len(), shorten(&s, 70));

    let mut x = Integer::from(100).pow(3000);
    x *= 2;
    let s = Integer::from(x.root(3)).to_string();
    println!("First {} digits of the cube root of 2:\n{}", s.len(), shorten(&s, 70));
}

Output:

Integer cube root of 8: 2
Integer cube root of 9: 2
First 2001 digits of the square root of 2:
14142135623730950488016887242096980...32952546758516447107578486024636008
First 2001 digits of the cube root of 2:
12599210498948731647672106072782283...28546452083111122546828353183047061

Scala

Functional solution, tail recursive, no immutables

import scala.annotation.tailrec

object IntegerRoots extends App {

  println("3rd integer root of 8 = " + iRoot(8, 3))

  println("3rd integer root of 9 = " + iRoot(9, 3))

  val result = iRoot(BigInt(100).pow(2000) * BigInt(2), 2)
  println(s"All ${result.toString.length} digits of the square root of 2: \n$result")

  private def iRoot(base: BigInt, degree: Int): BigInt = {
    require(base >= 0 && degree > 0,
      "Base has to be non-negative while the degree must be positive.")

    val (n1, n2) = (degree - 1, BigInt(degree))
    val d = (n1 + base) / n2

    @tailrec
    def loop(c: BigInt, d: BigInt, e: BigInt): BigInt = {
      if (c == d || c == e) if (d < e) d else e
      else loop(d, e, (n1 * e + (base / e.pow(n1))) / n2)
    }

    loop(1, (n1 + base) / n2, (n1 * d + (base / d.pow(n1))) / n2)
  }

}

Output:

See it running in your browser by ScalaFiddle (JavaScript, non JVM) or by Scastie (JVM).

Scheme

Translation of: Python

(define (root a b)
  (define // quotient)
  (define (y a a1 b c d e)
    (if (or (= c d) (= c e))
      (min d e)
      (y a a1 b d e (// (+ (* a1 e) (// b (expt e a1))) a))))
  (if (< b 2)
    b
    (let* ((a1 (- a 1))
           (c 1)
           (d (// (+ (* a1 c) (// b (expt c a1))) a))
           (e (// (+ (* a1 d) (// b (expt d a1))) a)))
      (y a a1 b c d e))))
      
(display "First 2,001 digits of the cube root of two:\n")      
(display (root 3 (* 2 (expt 1000 2000))))

Output:

First 2,001 digits of the cube root of two:
125992104989487316476721060727822835057025146470150798008197511215529967651395948372939656243625509415431025603561566525939902404061373722845911030426935524696064261662500097747452656548030686718540551868924587251676419937370969509838278316139915512931369536618394746344857657030311909589598474110598116290705359081647801147352132548477129788024220858205325797252666220266900566560819947156281764050606648267735726704194862076214429656942050793191724414809204482328401274703219642820812019057141889964599983175038018886895942020559220211547299738488026073636974178877921579846750995396300782609596242034832386601398573634339097371265279959919699683779131681681544288502796515292781076797140020406056748039385612517183570069079849963419762914740448345402697154762285131780206438780476493225790528984670858052862581300054293885607206097472230406313572349364584065759169169167270601244028967000010690810353138529027004150842323362398893864967821941498380270729571768128790014457462271477023483571519055067220848184850092872392092826466067171742477537097370300127429180940544256965920750363575703751896037074739934610144901451576359604711119738452991329657262589048609788561801386773836157730098659836608059757560127871214868562426845564116515581793532280158962912994450040120842541416015752584162988142309735821530604057724253836453253356595511725228557956227724036656284687590154306675351908548451181817520429124123378096317252135754114181146612736604578303605744026513096070968164006888185657231009008428452608641405950336900307918699355691335183428569382625543135589735445023330285314932245513412195545782119650083395771426685063328419619686512109255789558850899686190154670043896878665545309854505763765036008943306510356935777537249548436821370317162162183495809356208726009626785183418345652239744540004476021778894208183802786665306532663261864116007400747475473558527701689502063754132232329694243701742343491617690600723853902227681129777413872079823430391031628546452083111122546828353183047061

Sidef

Translation of: Ruby

func root(a, b) {
    b < 2 && return(b)
    var (a1, c) = (a-1, 1)
    var f = {|x| (a1*x + b//(x**a1)) // a }
    var d = f(c)
    var e = f(d)
    while (c !~ [d, e]) {
        (c, d, e) = (d, e, f(e))
    }
    [d, e].min
}

say "First 2,001 digits of the square root of two:"
say root(2, 2 * 100**2000)

Output:

First 2,001 digits of the square root of two:
14142135623730950488016887242096980[...]32952546758516447107578486024636008

Tcl

Tcl is not made for number crunching. The execution is quite slow compared to compiled languages.

On the other hand, everything is very straightforward, no libraries necessary.

proc root {this n} {
  if {$this < 2} {return $this}
  set n1 [expr $n - 1]
  set n2 $n
  set n3 $n1
  set c 1
  set d [expr ($n3 + $this) / $n2]
  set e [expr ($n3 * $d + $this / ($d ** $n1)) / $n2]
  while {$c != $d && $c != $e} {
    set c $d
    set d $e
    set e [expr ($n3 * $e + $this / ($e ** $n1)) / $n2]
  }
  return [expr min($d, $e)]
}

puts [root 8 3]
puts [root 9 3]
puts [root [expr 2* (100**2000)] 2]

Output:

2
2
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723528850926486124949771542183342042856860601468247207714358548741556570696776537202264854470158588016207584749226572260020855844665214583988939443709265918003113882464681570826301005948587040031864803421948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698683684507257993647290607629969413804756548237289971803268024744206292691248590521810044598421505911202494413417285314781058036033710773091828693147101711116839165817268894197587165821521282295184884720896946338628915628827659526351405422676532396946175112916024087155101351504553812875600526314680171274026539694702403005174953188629256313851881634780015693691768818523786840522878376293892143006558695686859645951555016447245098368960368873231143894155766510408839142923381132060524336294853170499157717562285497414389991880217624309652065642118273167262575395947172559346372386322614827426222086711558395999265211762526989175409881593486400834570851814722318142040704265090565323333984364578657967965192672923998753666172159825788602633636178274959942194037777536814262177387991945513972312740668983299898953867288228563786977496625199665835257761989393228453447356947949629521688914854925389047558288345260965240965428893945386466257449275563819644103169798330618520193793849400571563337205480685405758679996701213722394758214263065851322174088323829472876173936474678374319600015921888073478576172522118674904249773669292073110963697216089337086611567345853348332952546758516447107578486024636008

Visual Basic .NET

Library: System.Numerics

From the method described on the Wikipedia page. Included is an Integer Square Root function to compare results to the Integer Nth Square root function. One must choose the exponents carefully, otherwise one will obtain the digits of the nth root of 20, 200, 2000, etc..., instead of 2. For example, 4008 was chosen because it works for both n = 2 and n = 3, whereas 4004 was chosen for n = 7

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()) ' given example
        Console.WriteLine("Integer Cube Root of 9:")
        Console.WriteLine(IntNthRoot(3, 9).ToString()) ' given example
        Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, square root method):")
        Console.WriteLine(IntSqRoot(b).ToString()) ' reality check
        Console.WriteLine("Integer Square Root of 2, (actually 2 * 10 ^ 4008, nth root method):")
        Console.WriteLine(IntNthRoot(2, b).ToString()) ' given example
        Console.WriteLine("Integer Cube Root of 2, (actually 2 * 10 ^ 4008):")
        Console.WriteLine(IntNthRoot(3, b).ToString()) ' bonus example
        b /= 10000
        Console.WriteLine("Integer 7th Root of 2, (actually 2 * 10 ^ 4004):")
        Console.WriteLine(IntNthRoot(7, b).ToString()) ' bonus example
        If Diagnostics.Debugger.IsAttached Then Console.Read()
    End Sub

End Module

Output:

Integer Cube Root of 8:
2
Integer Cube Root of 9:
2
Integer Square Root of 2, (actually 2 * 10 ^ 4008, square root method):
1414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847208969463386289156288276595263514054226765323969461751129160240871551013515045538128756005263146801712740265396947024030051749531886292563138518816347800156936917688185237868405228783762938921430065586956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867115583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689832998989538672882285637869774966251996658352577619893932284534473569479496295216889148549253890475582883452609652409654288939453864662574492755638196441031697983306185201937938494005715633372054806854057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360083444
Integer Square Root of 2, (actually 2 * 10 ^ 4008, nth root method):
1414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989687253396546331808829640620615258352395054745750287759961729835575220337531857011354374603408498847160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237235288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162075847492265722600208558446652145839889394437092659180031138824646815708263010059485870400318648034219489727829064104507263688131373985525611732204024509122770022694112757362728049573810896750401836986836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112024944134172853147810580360337107730918286931471017111168391658172688941975871658215212822951848847208969463386289156288276595263514054226765323969461751129160240871551013515045538128756005263146801712740265396947024030051749531886292563138518816347800156936917688185237868405228783762938921430065586956868596459515550164472450983689603688732311438941557665104088391429233811320605243362948531704991577175622854974143899918802176243096520656421182731672625753959471725593463723863226148274262220867115583959992652117625269891754098815934864008345708518147223181420407042650905653233339843645786579679651926729239987536661721598257886026336361782749599421940377775368142621773879919455139723127406689832998989538672882285637869774966251996658352577619893932284534473569479496295216889148549253890475582883452609652409654288939453864662574492755638196441031697983306185201937938494005715633372054806854057586799967012137223947582142630658513221740883238294728761739364746783743196000159218880734785761725221186749042497736692920731109636972160893370866115673458533483329525467585164471075784860246360083445
Integer Cube Root of 2, (actually 2 * 10 ^ 4008):
12599210498948731647672106072782283505702514647015079800819751121552996765139594837293965624362550941543102560356156652593990240406137372284591103042693552469606426166250009774745265654803068671854055186892458725167641993737096950983827831613991551293136953661839474634485765703031190958959847411059811629070535908164780114735213254847712978802422085820532579725266622026690056656081994715628176405060664826773572670419486207621442965694205079319172441480920448232840127470321964282081201905714188996459998317503801888689594202055922021154729973848802607363697417887792157984675099539630078260959624203483238660139857363433909737126527995991969968377913168168154428850279651529278107679714002040605674803938561251718357006907984996341976291474044834540269715476228513178020643878047649322579052898467085805286258130005429388560720609747223040631357234936458406575916916916727060124402896700001069081035313852902700415084232336239889386496782194149838027072957176812879001445746227147702348357151905506722084818485009287239209282646606717174247753709737030012742918094054425696592075036357570375189603707473993461014490145157635960471111973845299132965726258904860978856180138677383615773009865983660805975756012787121486856242684556411651558179353228015896291299445004012084254141601575258416298814230973582153060405772425383645325335660
Integer 7th Root of 2, (actually 2 * 10 ^ 4004):
110408951367381233764950538762334472132532660078012416551453246414210632288038098071659828988630200514689715906557993125396921468043085579651064805838808196163919864392215583814551234397476339507890664685902921180613942144056283519219500774011043913929222338953790376732070503206390380988494445707084527925240582730725486467967183681658942999591682242459036160190261150569028438652686935172086652456800484770182207006433466758082204482396098451455092224240860882545144206285044829838431779372151867676523068340672781132725205233485925077681104722131036524174667129439905032

Wren

Wren doesn't have arbitrary precision numerics and so can't do the third example in the task description. We therefore do the third C/C++ example instead.

var intRoot = Fn.new { |x, n|
    if (!(x is Num && x.isInteger && x >= 0)) {
        Fiber.abort("First argument must be a non-negative integer.")
    }
    if (!(n is Num && x.isInteger && x >= 1)) {
        Fiber.abort("Second argument must be a positive integer.")
    }
    return x.pow(1/n).floor
}

var a = [ [8, 3], [9, 3], [2e18, 2] ]
for (e in a) {
    var x = e[0]
    var n = e[1]
    System.print("%(x) ^ (1/%(n)) = %(intRoot.call(x, n))")
}

Output:

8 ^ (1/3) = 2
9 ^ (1/3) = 2
2e+18 ^ (1/2) = 1414213562

Yabasic

Translation of: FreeBASIC

sub root(n, x)
    for nr = floor(sqr(x)) to 1 step -1
        if (nr ^ n) <= x then return nr : fi
    next nr
end sub

print root(3, 8) 
print root(3, 9) 
print root(4, 167)
end

zkl

Translation of: Python

Uses GNU GMP library

var [const] BN=Import("zklBigNum");
fcn root(n,r){
   f:='wrap(z){ (n/z.pow(r-1) + z*(r-1))/r or 1 };  //--> v or 1
   c,d,e:=1,f(c),f(d);
   while(c!=d and c!=e){ c,d,e=d,e,f(e) }
   if(d<e) d else e
}

a:=BN(100).pow(2000)*2;
println("Does GMP agree: ",root(a,3)==a.root(3));

Output:

Does GMP agree: True