# Exponentiation

## Exponentiation

Given two integers a and n, write a function to compute a^n.

#### Code

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}```int power(int x, unsigned int y) {
if (y == 0)
return 1;
else if (y%2 == 0)
return power(x, y/2)*power(x, y/2);
else
return x*power(x, y/2)*power(x, y/2);
} ``````

Time Complexity: O(n) | Space Complexity: O(1)

#### Optimized Solution: O(logn)

``````int power(int x, unsigned int y) {
int temp;
if( y == 0)
return 1;
temp = power(x, y/2);
if (y%2 == 0)
return temp*temp;
else
return x*temp*temp;
} ``````

Why is this faster?

Suppose we have x = 5, y = 4, we know that our answer is going to be (5 * 5 * 5 * 5).

If we break this down, we notice that we can write (5 * 5 * 5 * 5) as (5 * 5) * 2 and further, we can write (5 * 5) as 5 * 2.

Through this observation, we can optimize our function to O(log n) by calculating power(x, y/2) only once and storing it.

## Modular Exponentiation

Given three numbers x, y, and p, compute (x^y) % p

``````int power(int x, unsigned int y, int p) {
int res = 1;
x = x % p;
while (y > 0) {
if (y & 1)
res = (res*x) % p;

// y must be even now
y = y >> 1;
x = (x*x) % p;
}
return res;
} ``````

Time Complexity: O(Log y).