Pow(x, n)

Problem Statement:

Implement pow(x, n), which calculates x raised to the power n (i.e., xn).

• Example:

  Example 1:

  Input: x = 2.00000, n = 10
  Output: 1024.00000
  

  Example 2:

  Input: x = 2.10000, n = 3
  Output: 9.26100
  

  Example 3:

  Input: x = 2.00000, n = -2
  Output: 0.25000
  Explanation: 2-2 = 1/22 = 1/4 = 0.25

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✅ Solution: Binary Exponentiation (Iterative)

class Solution {
public:
    double myPow(double x, int n) {
        long long exp = n;  // Use long long to avoid overflow for INT_MIN
        if(exp < 0) {
            x = 1 / x;       // Take reciprocal if exponent is negative
            exp = -exp;
        }

        double result = 1;
        while(exp > 0){
            if(exp % 2 == 1)     // If exponent is odd, multiply result
                result *= x;
            x *= x;              // Square the base
            exp /= 2;            // Halve the exponent
        }
        return result;
    }
};

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📝 How It Works

• The function implements Binary Exponentiation to compute x^n in O(log n) time.
• It first handles the negative exponent case by taking the reciprocal (1/x) and flipping the sign.
• Then, using a loop, it squares the base x and halves the exponent n on each step.
• If the exponent is odd, it multiplies the result by current x (before squaring).
• This way, we reduce the number of multiplications from O(n) to O(log n).

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🧩 Key Formula / Recurrence

• Binary Exponentiation:

  x^n = (x^2)^(n/2)       if n is even
  x^n = x * (x^2)^(n/2)   if n is odd
  

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⏱️ Time & Space Complexity

Operation	Complexity
Time	O(log n)
Space	O(1)

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⚠️ Edge Cases

• n = 0: returns 1
• x = 0: returns 0 (except for 0^0, usually treated as 1)
• n = INT_MIN: handled using long long to avoid overflow when n
• Negative x with odd/even n: sign handled correctly

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💡 Other Approaches

Approach	Time	When to Use
Recursive Binary Exponentiation	O(log n)	Cleaner, uses recursion
Brute Force (x * x * ... * x)	O(n)	Only for small n (not recommended)

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🔁 Related Problems

• Leetcode 50. Pow(x, n)
• Implement integer exponentiation modulo m
• Matrix exponentiation (T^n)
• Fast modular exponentiation (x^n % m)

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