# N-th root of a number

Given two numbers N and A, find N-th root of A. In mathematics, Nth root of a number A is a real number that gives A, when we raise it to integer power N. These roots are used in Number Theory and other advanced branches of mathematics.
Examples:

```Input : A = 81
N = 4
Output : 3
3^4 = 81
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

As this problem involves a real valued function A^(1/N) we can solve this using Newton’s method, which starts with an initial guess and iteratively shift towards the result.
We can derive a relation between two consecutive values of iteration using Newton’s method as follows,

```according to newton’s method
x(K+1) = x(K) – f(x) / f’(x)
here    f(x)  = x^(N) – A
so    f’(x) = N*x^(N - 1)
and     x(K) denoted the value of x at Kth iteration
putting the values and simplifying we get,
x(K + 1) = (1 / N) * ((N - 1) * x(K) + A / x(K) ^ (N - 1))
```

Using above relation, we can solve the given problem. In below code we iterate over values of x, until difference between two consecutive values of x become lower than desired accuracy.

## C++

```// C++ program to calculate Nth root of a number
#include <bits/stdc++.h>
using namespace std;

//  method returns Nth power of A
double nthRoot(int A, int N)
{
// intially guessing a random number between
// 0 and 9
double xPre = rand() % 10;

//  smaller eps, denotes more accuracy
double eps = 1e-3;

// initializing difference between two
// roots by INT_MAX
double delX = INT_MAX;

//  xK denotes current value of x
double xK;

//  loop untill we reach desired accuracy
while (delX > eps)
{
//  calculating current value from previous
// value by newton's method
xK = ((N - 1.0) * xPre +
(double)A/pow(xPre, N-1)) / (double)N;
delX = abs(xK - xPre);
xPre = xK;
}

return xK;
}

//    Driver code to test above methods
int main()
{
int N = 4;
int A = 81;

double nthRootValue = nthRoot(A, N);
cout << "Nth root is " << nthRootValue << endl;

/*
double Acalc = pow(nthRootValue, N);
cout << "Error in difference of powers "
<< abs(A - Acalc) << endl;
*/

return 0;
}
```

## Java

```// Java program to calculate Nth root of a number
class GFG
{

// method returns Nth power of A
static double nthRoot(int A, int N)
{

// intially guessing a random number between
// 0 and 9
double xPre = Math.random() % 10;

// smaller eps, denotes more accuracy
double eps = 0.001;

// initializing difference between two
// roots by INT_MAX
double delX = 2147483647;

// xK denotes current value of x
double xK = 0.0;

// loop untill we reach desired accuracy
while (delX > eps)
{
// calculating current value from previous
// value by newton's method
xK = ((N - 1.0) * xPre +
(double)A / Math.pow(xPre, N - 1)) / (double)N;
delX = Math.abs(xK - xPre);
xPre = xK;
}

return xK;
}

// Driver code
public static void main (String[] args)
{
int N = 4;
int A = 81;

double nthRootValue = nthRoot(A, N);
System.out.println("Nth root is "
+ Math.round(nthRootValue*1000.0)/1000.0);

/*
double Acalc = pow(nthRootValue, N);
cout << "Error in difference of powers "
<< abs(A - Acalc) << endl;
*/
}
}

// This code is contributed by Anant Agarwal.
```

## Python3

```# Python3 program to calculate
# Nth root of a number
import math
import random

# method returns Nth power of A
def nthRoot(A,N):

# initially guessing a random number between
# 0 and 9
xPre = random.randint(1,101) % 10

#  smaller eps, denotes more accuracy
eps = 0.001

# initializing difference between two
# roots by INT_MAX
delX = 2147483647

#  xK denotes current value of x
xK=0.0

#  loop untill we reach desired accuracy
while (delX > eps):

# calculating current value from previous
# value by newton's method
xK = ((N - 1.0) * xPre +
A/pow(xPre, N-1)) /N
delX = abs(xK - xPre)
xPre = xK;

return xK

# Driver code
N = 4
A = 81
nthRootValue = nthRoot(A, N)

print("Nth root is ", nthRootValue)

## Acalc = pow(nthRootValue, N);
## print("Error in difference of powers ",
##             abs(A - Acalc))

# This code is contributed
# by Anant Agarwal.
```

Output:

```Nth root is 3
```

This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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