# Find Nth root of a number using Bisection method

• Last Updated : 28 Jul, 2022

Given two positive integers N and P. The task is to find the Nth root of P.

Examples:

Input: P = 1234321, N = 2
Output: 1111
Explanation: square root of 1234321 is 1111.

Input: P = 123456785, N = 20
Output: 2.53849

Approach: There are various ways to solve the given problem. Here the below algorithm is based on Mathematical Concept called Bisection Method for finding roots. To find the N-th power root of a given number P we will form an equation is formed in x as  ( xp – P = 0 ) and the target is to find the positive root of this equation using the Bisection Method.

How Bisection Method works?
Take an interval (a, b) such that its already known that the root is existing in that interval. After this find the mid of the interval and examine the value of function and it’s derivative at x = mid.

• If the value of function is 0 that means root is found
• If the value of the function is positive and its derivative is negative that means the root is lying in the right half.
• If the value of the function is positive and its derivative is positive that means the root is lying in the left half.

Below is the implementation of the above approach:

## C++14

 `// C++ program for above approach``#include ``using` `namespace` `std;` `// Function that returns the value``// of the function at a given value of x``double` `f(``double` `x, ``int` `p, ``double` `num)``{``    ``return` `pow``(x, p) - num;``}``// calculating the value``// of the differential of the function``double` `f_prime(``double` `x, ``int` `p)``{``    ``return` `p * ``pow``(x, p - 1);``}` `// The function that returns``// the root of given number``double` `root(``double` `num, ``int` `p)``{` `    ``// Defining range``    ``// on which answer can be found``    ``double` `left = -num;``    ``double` `right = num;` `    ``double` `x;``    ``while` `(``true``) {` `        ``// finding mid value``        ``x = (left + right) / 2.0;``        ``double` `value = f(x, p, num);``        ``double` `prime = f_prime(x, p);``        ``if` `(value * prime <= 0)``            ``left = x;``        ``else``            ``right = x;``        ``if` `(value < 0.000001 && value >= 0) {``            ``return` `x;``        ``}``    ``}``}` `// Driver code``int` `main()``{` `    ``double` `P = 1234321;``    ``int` `N = 2;` `    ``double` `ans = root(P, N);``    ``cout << ans;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;` `class` `GFG``{``  ` `  ``// Function that returns the value``  ``// of the function at a given value of x``  ``static` `double` `f(``double` `x, ``int` `p, ``double` `num)``  ``{``      ``return` `Math.pow(x, p) - num;``  ``}``  ` `  ``// calculating the value``  ``// of the differential of the function``  ``static` `double` `f_prime(``double` `x, ``int` `p)``  ``{``      ``return` `p * Math.pow(x, p - ``1``);``  ``}` `  ``// The function that returns``  ``// the root of given number``  ``static` `double` `root(``double` `num, ``int` `p)``  ``{` `      ``// Defining range``      ``// on which answer can be found``      ``double` `left = -num;``      ``double` `right = num;` `      ``double` `x;``      ``while` `(``true``) {` `          ``// finding mid value``          ``x = (left + right) / ``2.0``;``          ``double` `value = f(x, p, num);``          ``double` `prime = f_prime(x, p);``          ``if` `(value * prime <= ``0``)``              ``left = x;``          ``else``              ``right = x;``          ``if` `(value < ``0.000001` `&& value >= ``0``) {``              ``return` `x;``          ``}``      ``}``  ``}` `  ``// Driver code``  ``public` `static` `void` `main(String args[])``  ``{` `      ``double` `P = ``1234321``;``      ``int` `N = ``2``;` `      ``double` `ans = root(P, N);``      ``System.out.print(ans);``  ``}``}` `// This code is contributed by ihritik`

## Python3

 `# python program for above approach` `# Function that returns the value``# of the function at a given value of x``def` `f(x, p, num):` `    ``return` `pow``(x, p) ``-` `num` `# calculating the value``# of the differential of the function``def` `f_prime(x, p):` `    ``return` `p ``*` `pow``(x, p ``-` `1``)` `# The function that returns``# the root of given number``def` `root(num, p):` `    ``# Defining range``    ``# on which answer can be found``    ``left ``=` `-``num``    ``right ``=` `num` `    ``x ``=` `0``    ``while` `(``True``):` `        ``# finding mid value``        ``x ``=` `(left ``+` `right) ``/` `2.0``        ``value ``=` `f(x, p, num)``        ``prime ``=` `f_prime(x, p)``        ``if` `(value ``*` `prime <``=` `0``):``            ``left ``=` `x``        ``else``:``            ``right ``=` `x``        ``if` `(value < ``0.000001` `and` `value >``=` `0``):``            ``return` `x` `# Driver code``if` `__name__ ``=``=` `"__main__"``:` `    ``P ``=` `1234321``    ``N ``=` `2` `    ``ans ``=` `root(P, N)``    ``print``(ans)` `# This code is contributed by rakeshsahni`

## C#

 `// C# program for the above approach` `using` `System;` `public` `class` `GFG``{``  ` `  ``// Function that returns the value``  ``// of the function at a given value of x``  ``static` `double` `f(``double` `x, ``int` `p, ``double` `num)``  ``{``      ``return` `Math.Pow(x, p) - num;``  ``}``  ` `  ``// calculating the value``  ``// of the differential of the function``  ``static` `double` `f_prime(``double` `x, ``int` `p)``  ``{``      ``return` `p * Math.Pow(x, p - 1);``  ``}` `  ``// The function that returns``  ``// the root of given number``  ``static` `double` `root(``double` `num, ``int` `p)``  ``{` `      ``// Defining range``      ``// on which answer can be found``      ``double` `left = -num;``      ``double` `right = num;` `      ``double` `x;``      ``while` `(``true``) {` `          ``// finding mid value``          ``x = (left + right) / 2.0;``          ``double` `value = f(x, p, num);``          ``double` `prime = f_prime(x, p);``          ``if` `(value * prime <= 0)``              ``left = x;``          ``else``              ``right = x;``          ``if` `(value < 0.000001 && value >= 0) {``              ``return` `x;``          ``}``      ``}``  ``}` `  ``// Driver code``  ``public` `static` `void` `Main(``string` `[]args)``  ``{` `      ``double` `P = 1234321;``      ``int` `N = 2;` `      ``double` `ans = root(P, N);``      ``Console.Write(ans);``  ``}``}` `// This code is contributed by AnkThon`

## Javascript

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Output

`1111`

Time Complexity: O(log P).
Auxiliary Space: O(1).

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