Power of a Number (Fast Exponentiation)
A medium Recursion problem included in Apna College, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Recursion
- Sheets
- 2
- Core for
- 11 roles
- Platform
- LeetCode
The problem
Implement the power function to compute x raised to the power n using fast exponentiation.
Example 1
- Input
- x = 2.0, n = 10
- Output
- 1024.0
- Why
- 2^10 = 1024.
Example 2
- Input
- x = 2.1, n = 3
- Output
- 9.261
Constraints
- -100.0 < x < 100.0
- -2^31 <= n <= 2^31 - 1
How to think about it
Updated 2026-09-09Multiplying by x one factor at a time is redundant when you can square the base. Every bit in the binary representation of n represents a power-of-two exponent. Whenever n is even, halving the exponent and squaring the base halves the remaining work without changing the mathematical value.
Approaches, worst first
Linear multiplication
time O(n) · space O(1)
Multiply 1.0 by x a total of n times. Unusable for large n because 2^31 iterations exceeds typical time limits by orders of magnitude.
Recursive binary exponentiation
time O(log n) · space O(log n)
Branch on whether n is odd or even: x^n = x * x^(n-1) or (x * x)^(n/2). Slashes the exponent in half each step, but consumes call stack frames proportional to log n.
Iterative binary exponentiationWrite this one
time O(log n) · space O(1)
Inspect the lowest bit of n. If set, multiply the running result by current base; then square the base and shift n right. Runs in logarithmic time with zero call stack overhead.
Where people lose marks · 3
- Negating n when n = -2^31. In 32-bit signed two's complement, -(-2^31) overflows back to -2^31, causing an infinite loop unless promoted to a 64-bit integer before inversion.
- Forgetting to invert the base or result when n is negative. x^(-n) is (1 / x)^n, not -x^n.
- Handling x = 0 with negative n, though the problem constraints typically guarantee non-zero base for negative powers to avoid division by zero.
The theory behind it
Recursion — the ground this problem stands on. All Recursion problems
What Recursion is
Recursion is a nesting doll that opens to reveal an identical smaller doll inside. In programming, a function solves a substantial problem by delegating smaller versions of the exact same question to fresh invocations of itself. Each invocation operates on shrunken input until hitting an irreducible foundation called a base case, which returns an immediate answer and permits the waiting cascade to resolve backwards.
When to reach for it
Reach for recursion when a problem possesses self-similar subproblems, such as traversing branched tree structures, exploring graph pathways, or generating combinations. Phrases asking for all permutations, subset generation, exhaustive maze navigation, or hierarchical file system traversals signal recursive decomposition. It is natural whenever the answer to a large instance depends on assembling identical solutions for smaller subsets.
How the pattern works
Structure every recursive method around two mandatory stages: the termination stop and the shrinking recurrence. Write the base condition first so the function exits before attempting further execution. Next, trust the recursive call to return valid answers for smaller inputs without mentally unwinding every level at once. Pass accumulation state forward through parameters, or combine child return values on the ascent phase once deeper calls return.
What each operation costs
| Operation | Time |
|---|---|
| call stack memory allocation per frame | O(d) |
| traversal of branching recursive call tree | O(b^d) |
| single branch linear recursive unwind | O(n) |
What usually goes wrong with Recursion
- Omitting a base case or writing a condition that input values leap over without triggering, triggering fatal call stack overflow crashes.
- Modifying shared mutable containers across sibling branches without undoing edits on backtracking steps, contaminating alternative search paths.
- Recomputing duplicate subproblems inside branching calls without memoizing past returns, causing execution times to explode exponentially.
Which roles need this problem
Recursion is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 5 more roles, including SDE / Backend Engineer, Full-Stack Developer, Security Engineer.
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