Given two number x and n, find n-th root of x.
Examples:

Input : 5 2
Output : 2.2360679768025875

Input :  x = 5, n = 3
Output : 1.70997594668

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

In order to calculate nth root of a number, we can use the following procedure.

If x lies in the range [0, 1) then we set the lower limit low = x and upper limit high = 1, because for this range of numbers the nth root is always greater than the given number and can never exceed 1.
eg- $$\sqrt{0.09} = 0.3$.$
Otherwise, we take low = 1 and high = x.
Declare a variable named epsilon and initialize it for accuracy you need.
Say epsilon=0.01, then we can guarantee that our guess for nth root of the given number will be
correct up to 2 decimal places.
Declare a variable guess and initialize it to guess=(low+high)/2.
Run a loop such that:
- if the absolute error of our guess is more than epsilon then do:
  1. if guessⁿ > x, then high=x
  2. else low=x
  3. Making a new better guess i.e., guess=(low+high)/2.
- If the absolute error of our guess is less than epsilon then exit the loop.

Absolute Error: Absolute Error can be calculated as abs(guessⁿ -x)

C++

// C++ Program to find  
// n-th real root of x 
#include<bits/stdc++.h> 
using namespace std; 
  
void findNthRoot(double x, int n) 
{ 
  
// Initialize boundary values  
double low, high; 
if (x >= 0 and x <= 1) 
{ 
    low = x; 
    high = 1; 
} 
else
{ 
    low = 1; 
    high = x;  
}  
  
// used for taking approximations  
// of the answer  
double epsilon = 0.00000001; 
  
// Do binary search  
double guess = (low + high) / 2; 
while (abs((pow(guess, n)) - x) >= epsilon) 
{ 
    if (pow(guess, n) > x) 
    { 
        high = guess ; 
    }  
    else
    { 
        low = guess ; 
    } 
    guess = (low + high) / 2; 
} 
  
cout << fixed << setprecision(16)  
     << guess; 
}      
  
// Driver code  
int main() 
{ 
    double x = 5; 
    int n = 2; 
    findNthRoot(x, n) ; 
} 
  
// This code is contributed 
// by Subhadeep 

Java

// Java Program to find n-th real root of x 
class GFG 
{ 
static void findNthRoot(double x, int n) 
{ 
  
// Initialize boundary values  
double low, high; 
if (x >= 0 && x <= 1) 
{ 
    low = x; 
    high = 1; 
} 
else
{ 
    low = 1; 
    high = x;  
}  
  
// used for taking approximations  
// of the answer  
double epsilon = 0.00000001; 
  
// Do binary search  
double guess = (low + high) / 2; 
while (Math.abs((Math.pow(guess, n)) - x) >= epsilon) 
{ 
    if (Math.pow(guess, n) > x) 
    { 
        high = guess ; 
    }  
    else
    { 
        low = guess ; 
    } 
    guess = (low + high) / 2; 
} 
  
System.out.println(guess); 
}  
  
// Driver code  
public static void main(String[] args) 
{ 
    double x = 5; 
    int n = 2; 
    findNthRoot(x, n) ; 
} 
} 
  
// This code is contributed 
// by mits 

Python3

# Python Program to find n-th real root  
# of x 
  
def findNthRoot(x, n): 
  
    # Initialize boundary values 
    x = float(x) 
    n = int(n) 
    if (x >= 0 and x <= 1): 
        low = x 
        high = 1
    else: 
        low = 1
        high = x 
         
    # used for taking approximations  
    # of the answer    
    epsilon = 0.00000001
  
    # Do binary search 
    guess = (low + high) / 2
    while abs(guess ** n - x) >= epsilon: 
        if guess ** n > x: 
            high = guess 
        else: 
            low = guess 
        guess = ( low + high ) / 2
    print(guess) 
      
  
# Driver code     
x = 5
n = 2
findNthRoot(x, n) 

C#

// C# Program to find n-th real root of x 
  
using System; 
  
public class GFG{ 
static void findNthRoot(double x, int n) 
{ 
  
// Initialize boundary values  
double low, high; 
if (x >= 0 && x <= 1) 
{ 
    low = x; 
    high = 1; 
} 
else
{ 
    low = 1; 
    high = x;  
}  
  
// used for taking approximations  
// of the answer  
double epsilon = 0.00000001; 
  
// Do binary search  
double guess = (low + high) / 2; 
while (Math.Abs((Math.Pow(guess, n)) - x) >= epsilon) 
{ 
    if (Math.Pow(guess, n) > x) 
    { 
        high = guess ; 
    }  
    else
    { 
        low = guess ; 
    } 
    guess = (low + high) / 2; 
} 
  
Console.WriteLine(guess); 
}  
  
// Driver code  
    static public void Main (){ 
    double x = 5; 
    int n = 2; 
    findNthRoot(x, n) ; 
} 
} 
  
// This code is contributed by akt_mit 

Output:

2.2360679768025875

Explanation of first example with epsilon = 0.01

Since taking too small value of epsilon as taken in our program might not be feasible for
explanation because it will increase the number of steps drastically so for the sake of
simplicity we are taking epsilon = 0.01
The above procedure will work as follows:
Say we have to calculate the then x = 5, low = 1, high = 5.
Taking epsilon = 0.01
First Guess:
guess = (1 + 5) / 2 = 3
Absolute error = |3² - 5| = 4 > epsilon
guess² = 9 > 5(x) then high = guess --> high = 3

Second Guess:
guess = (1 + 3) / 2 = 2
Absolute error = |2² - 5| = 1 > epsilon
guess² = 4 > 5(x) then low = guess --> low = 2

Third Guess:
guess = (2 + 3) / 2 = 2.5
Absolute error = |2.5² - 5| = 1.25 > epsilon
guess² = 6.25 > 5(x) then high = guess --> high = 2.5
and proceeding so on we will get the  correct up to 2 decimal places i.e.,  = 2.23600456
We will ignore the digits after 2 decimal places since they may or may not be correct.

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My Personal Notes

Calculating n-th real root using binary search

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

Explanation of first example with epsilon = 0.01

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