Question

(Recursive power Method) Write a recursive method power(base, exponent) that, when called, returns base exponent For example, power \((3,4)=3^{*} 3^{*} 3^{*} 3 .\) Assume that exponent is an integer greater than or equal to 1\. Hint: The recursion step should use the relationship base exponent \(=\) base \(\cdot\) base exponent- 1 and the terminating condition occurs when exponent is equal to 1 , because base \(^{l}=\) base Incorporate this method into a program that enables the user to enter the base and exponent.

Short answer

To compute base raised to the exponent recursively, define a function power(base, exponent) that returns base if exponent is 1 (base case), or base multiplied by power(base, exponent - 1) for the recursive case.

Step-by-step solution

Understanding the Problem

First, you need to understand that the task is to create a recursive function called power(...), which takes two parameters: base and exponent. The function should return the value of the base raised to the power of the exponent. Recursion means the function will call itself with a simpler problem until it reaches the terminating condition.

Writing the Base Case

In any recursive function, there has to be a base case that stops the recursion. For this exercise, the base case occurs when the exponent is equal to 1. According to the given hint, if exponent is 1, the function should return base, because any number raised to the power of 1 is itself.

Writing the Recursive Case

The recursive step is where the function calls itself. The relationship base exponent = base * base exponent-1 should be used to create the recursive step. This means you multiply the base by the result of a recursive call to the power(...) function with the exponent decreased by 1.

Combining the Steps into a Function

Now, create the power(base, exponent) function using the base and recursive cases. If the exponent is not 1, the function should call itself with exponent decremented by 1 and multiply the result by base. Otherwise, it just returns the base.

Creating a User Interface

To incorporate this function into a program, provide a way for the user to input the base and exponent values, for example through a text field or console input. After getting the input, call the power(...) function with these values and display the result.

Testing the Function

Finally, test the function with various base and exponent pairs to ensure it works correctly. Verify if the output matches the expected result using either manual calculations or a calculator.

Key concepts

Recursion in Java

Recursion is a powerful concept in programming, particularly useful when solving problems that can be broken down into smaller instances of the same problem. In Java, recursion is achieved when a method calls itself within its own code. However, it's crucial to manage this process carefully to avoid an infinite loop, which could result in a stack overflow error.

Consider a task to handle repetitive actions, like walking down a flight of steps. You could describe it recursively as: if there's still a step ahead, move down one step and repeat the process. Once there's no step left, you stop, which is similar to reaching the 'base case' in a recursive function.

To implement recursion for the power method in Java, follow these guidelines:

• Define a method that takes parameters relevant to the problem, such as 'base' and 'exponent'.
• Write the recursive logic inside the method, ensuring there's a condition that will lead to the method eventually stopping the recursive calls (the base case).
• Test the method for various inputs to ensure the recursion stops as expected and provides the correct result.

Base Case in Recursion

The base case in recursion is the scenario where the recursive calls stop. It is a critical element of a recursive function, acting as a terminating condition that prevents infinite recursion. In our power method example, the base case occurs when the exponent is 1. At this point, we know that any number to the power of one is the number itself, making further recursion unnecessary.

In a recursive function, the base case should always be checked first, before any recursive calls are made. It typically involves a simple condition or comparison. The goal is to identify the simplest version of the problem, which can be solved without further recursion.

Here's how to define a base case effectively:

• Identify the simplest instance of the problem that doesn't require recursion.
• Create a condition in your recursive method that checks for this instance before proceeding with recursive calls.
• Ensure the base case returns a value, which will be used in the previous iterations of the recursion to eventually solve the entire problem.

Exponentiation Algorithm

An exponentiation algorithm is a method for finding the power of a number, that is, calculating the result of raising a base to a certain exponent. Recursive exponentiation harnesses the principle of divide and conquer; it simplifies the problem into smaller parts, resolving them recursively and combining their results to find the final answer.

The algorithm for recursive exponentiation based on our problem can be outlined as follows:

• If the exponent is 1, return the base (as that's the base case).
• Otherwise, recursively call the power function with the exponent decreased by 1 and multiply this result by the base to get the intermediate result.
• Repeat this process until the base case is encountered.

In this way, the function power(base, exponent) effectively uses the relationship \( base^{exponent} = base \times base^{exponent-1} \) to calculate the result.

By building each layer of the recursion on top of the previous one, the algorithm expedites the calculation and efficiently reduces the complexity of the operation. This is a practical application of recursion and illustrates how it can be used to implement mathematical operations in a programmable way.