Question

Write a recursive function power( base, exponent ) that, when invoked, returns base \(^{exponent}\) For example, power \((3,4)=3 \pm 3 * 3 * 3 .\) Assume that exponent is an integer greater than or equal to 1\. Hint: The recursion step would use the relationship base \(^{exponent}\) \(=\) base \(\cdot\) base \(^{exponent - 1}\) and the terminating condition occurs when exponent is equal to \(1,\) because base \(^{1}=\) base

Short answer

The recursive function 'power(base, exponent)' returns 'base' raised to 'exponent' by recursively multiplying 'base' by 'power(base, exponent - 1)', until 'exponent' equals 1.

Step-by-step solution

Understanding the Problem

We are to write a recursive function named 'power' that takes two arguments: 'base' and 'exponent'. The function should return the 'base' raised to the power of 'exponent'. The recursive relationship we'll use is that base ^ exponent equals base times base ^ (exponent - 1). The base case occurs when exponent is equal to 1, at which point the function should return the base itself.

Defining the Base Case

We start by defining the base case of our recursive function. The base case occurs when the exponent is 1. In this situation, the function should simply return the base because any number raised to the power of 1 is itself.

Defining the Recursive Case

For the recursive case, we call the 'power' function with the same 'base' and 'exponent' decremented by 1. The function should return the result of multiplying the 'base' by the result of 'power(base, exponent - 1)'.

Writing the Recursive Function

We write the 'power' function by combining the base case and the recursive case in a function definition. If the exponent is 1, return base. Otherwise, recursively multiply the base by the power function of the base and exponent - 1.

Testing the Function

Finally, we can test our function by calling it with sample arguments to ensure it behaves as expected, for instance, calculating power(3,4) and checking if it equals 81.

Key concepts

Base Case in Recursion

The concept of a base case is central to understanding recursion in programming. Imagine a base case as the stopping point of the recursion, the condition under which the function will stop calling itself and begin to return values.

In our exercise, the base case occurs when the exponent is 1. This is a critical step in the design of a recursive algorithm, as it defines a simple, solvable instance of the problem for which the answer is known and can be returned directly without further recursion. For the function power(base, exponent), when exponent is 1, the function will simply return base, because mathematically, any number to the power of one is itself (\( base^1 = base \)). Properly identifying the base case is crucial, because without it, the recursion could lead to an infinite loop, causing a stack overflow error.

Recursive Relationship

The recursive relationship is the rule that breaks down the complex problem into simpler sub-problems by calling the function within itself with modified arguments. In recursion, this self-referential step is essential as it approaches the problem incrementally towards the base case.

In the context of our power function, the recursive relationship is established by the expression \( base^{exponent} = base \times base^{exponent - 1} \). With each recursive call, the exponent is decremented by one, which gradually moves the calculation closer to the base case. As each call waits for the next call to complete, they stack up creating a chain of unfinished operations, often visualized with a call stack. Once the base case is reached and the value is returned, the stack unwinds, completing each deferred operation sequentially until the initial call is resolved and the final result is returned.

Exponentiation Algorithm

The exponentiation algorithm we've discussed implements an efficient method to calculate the power of a number. Unlike iterative approaches that may use repeated multiplication, our recursive method efficiently reduces the problem size with each function call.

To understand the algorithm with our function power(base, exponent), let's consider an example: power(3, 4). This will lead to multiple calls: power(3, 3), power(3, 2), and finally power(3, 1). The last call returns 3 (the base case), and then the previous calls complete, eventually calculating 3 * 3 * 3 * 3, or 81.

This method highlights a divide-and-conquer strategy, where the larger problem is divided into smaller ones that are easier to manage and solve. By focusing on the recursive relationship and the base case, it's possible to implement an effective and clear solution to exponentiation without resorting to loops or iterative logic, which is particularly useful in languages like C++ that support recursion natively.