Question

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 cqual to 1. \([\text {Hint: }\) The recursion step should use the relationship base \(^{exponent}\) = base base \(^{exponent - l}\) and the terminating condition occurs when exponent is equal to \(1,\) because base \(^{l}=\) base

Short answer

The recursive method returns base when exponent is 1; otherwise, it returns base multiplied by power(base, exponent - 1).

Step-by-step solution

- Define the Base Case

Start by specifying the base case for the recursive method. The base case occurs when the exponent is equal to 1 because any number raised to the power of 1 is the number itself. In this case, if exponent is 1, the method should return the base.

- Define the Recursion Step

The recursive step will involve calling the power method itself with a reduced exponent. According to the recursive relation, base raised to the power of exponent is the same as base times base raised to the power of exponent minus 1. Thus, the method should return the product of base and the method power(base, exponent - 1).

- Combine Base Case and Recursive Step

Implement the method by combining the base case and the recursion step. If exponent is 1, return base; otherwise, return base * power(base, exponent - 1). This method will continue to call itself with a decremented exponent until the base case is reached.

- Test the Method

Test the recursive method with various examples to ensure it works correctly. For instance, calling power(3, 4) should return 81, as 3 multiplied by itself 4 times is 81 (3 * 3 * 3 * 3).

Key concepts

Recursion

Recursion is a fundamental programming concept where a function calls itself to solve smaller instances of the same problem. In essence, it breaks down a complex problem into simpler, more manageable sub-problems until it reaches a point where the problem can be solved without further recursion.

For a recursive method to work properly, it must have a specified 'base case,' which is a condition under which the recursion ends and the method no longer calls itself. Without this, recursion would lead to an infinite loop, potentially causing a stack overflow error. Recursive methods should also ensure that each recursive call progressively leads to the base case, thereby making the problem smaller each time.

Applying recursion to calculate powers of a number offers a clear example. The problem of calculating the power function can be reduced repeatedly by breaking down the exponent term until it reaches the base case. Recursion provides a neat and elegant solution compared to iterative methods, especially for mathematical functions.

Power Function

The power function is used to raise a number, known as the base, to an exponent, which is the power it is raised to. The function is an important concept in mathematics and is extensively used in various fields, including science and engineering.

In programming, implementing the power function using recursion brings clarity and simplicity to the code. The power function can be expressed as a series of multiplications, and this repetitive process is a perfect fit for a recursive approach. This is because the power function itself can be broken down into multiplying the base with the power of the base to the decrement of the exponent until it reaches the known result of any number to the power of 1.

Base and Exponent

The base and exponent are two parts of an expression that denotes repeated multiplication. The base is the number that is being multiplied by itself, and the exponent tells us how many times the base is used as a factor in the multiplication.

For instance, in the expression base^exponent, if the base is 3 and the exponent is 4, it represents the number 3 being multiplied by itself 3 times: 3 * 3 * 3 * 3. Understanding this concept is essential when dealing with powers and roots in both mathematics and programming. It is the exponent that recursion reduces in each call in the recursive method power(base, exponent) - by subtracting one until it reaches the base case.

Programming Concepts

Programming concepts such as variables, control structures, data structures, syntax, and recursion are tools that developers use to build software. These concepts form the foundation of writing any computer program, allowing developers to solve complex problems and execute logical and mathematical operations efficiently.

In case of the power function, the variables are the base and exponent, the control structure is the conditional statement that determines whether to continuing the recursion or return the result based on the base case, and the data structure might be as simple as the numerical data types used to hold the values of the base and exponent.

Writing code to perform the power operation recursively also provides an opportunity to practice debugging and testing - key parts of software development. As each recursive call is made, the state of the program changes and must be managed correctly. Tests, such as ensuring the method handles various values for base and exponent, are integral in verifying that the recursion behaves as expected.