Question

Redefine the function power so that it also works for negative exponents. In order to do this you will also have to change the type of the value returned to double. The function declaration and header comment for the redefined version of power is as follows: double power \((\text { int } x, \text { int } n)\) //Precondition: If \(n<\theta\), then \(x\) is not \(\theta\) //Returns \(x\) to the power \(n\) Hint: \(x^{-\mathrm{n}}\) is equal to \(1 /\left(x^{\mathrm{n}}\right)\)

Short answer

Question: Redefine the function "power(x, n)" to work for negative exponents as well, keeping in mind that x^(-n) is equal to 1/(x^n). Answer: To redefine the function "power(x, n)" to work for negative exponents, we can modify it as follows: ```c double power(int x, int n) { double result = 1.0; if (n < 0) { n = -n; for (int i = 0; i < n; i++) { result *= x; } result = 1 / result; } else { for (int i = 0; i < n; i++) { result *= x; } } return result; } ``` This function checks if the exponent 'n' is negative; if it is, it changes the sign of 'n' and calculates the result as before. Then, it divides 1 by the result in order to handle the negative exponent case.

Step-by-step solution

Write the function declaration and header comment

Begin by writing the function declaration as follows: ```c double power(int x, int n) ``` And include the header comments given: ```c /* Precondition: If n < 0, then x is not 0 Returns x to the power n */ ```

Implement the function for positive exponents

If the exponent (n) is positive, we can simply multiply the base (x) by itself n times. First, declare a variable to store the result and initialize it to 1, since any number raised to 0 is 1: ``` double result = 1.0; ``` Next, create a for loop that iterates 'n' times: ``` for (int i = 0; i < n; i++) { result *= x; } ```

Handle the case for negative exponents

To handle negative exponents, we will first check if 'n' is negative. If it is, we will simply change the sign of 'n' and multiply the base in the same manner as before, then divide 1 by the result as the hint suggests (\(x^{-n} = \frac{1}{x^n}\)): ```c if (n < 0) { n = -n; for (int i = 0; i < n; i++) { result *= x; } result = 1 / result; } ```

Return the result

Finally, after calculating the result, we will return it: ```c return result; ```

Full Redefined Power Function

The complete redefined power function should look like this: ```c double power(int x, int n) { /* Precondition: If n < 0, then x is not 0 Returns x to the power n */ double result = 1.0; if (n < 0) { n = -n; for (int i = 0; i < n; i++) { result *= x; } result = 1 / result; } else { for (int i = 0; i < n; i++) { result *= x; } } return result; } ```

Key concepts

Negative Exponents Handling

Handling negative exponents can be tricky since multiplying involves division when an exponent is negative. In mathematics, raising a number to a negative exponent means finding the reciprocal of the number raised to the opposite positive exponent. For example, raising 2 to -3 means you compute \[2^{-3} = \frac{1}{2^3} = \frac{1}{8}\].In C++, we handle negative exponents by first converting the exponent to positive. We multiply the base by itself that many times, just like we would for positive exponents. Once we have the result, we take the reciprocal of that result to account for the negative exponent:- Check if the exponent is negative.- Convert negative exponents to positive: `n = -n`.- Multiply base `x` by itself `n` times.- Take reciprocal: `result = 1 / result`.By doing this, our function correctly handles negative exponents by effectively performing mathematical division.

Return Type Change

In order to handle negative exponents and fractions, the return type of the function must change from int to double. Double precision floating-point provides us the ability to represent numerical values that have a decimal component. This is essential when dealing with number operations that involve division, as results are often not whole numbers, especially when dealing with fractions.

• Using a double return type allows us to represent values like 0.125 without rounding errors.
• It's important to remember to initialize the result variable as a double too, like `double result = 1.0;`.

This datatype change ensures that our program can safely handle and return precise results from our calculations involving negative exponents.

Algorithm Implementation Steps

Implementing the function to handle both positive and negative exponents involves several steps. Each step is crucial for ensuring that the power function works correctly for all integer inputs. Here's a breakdown of the setup: - **Start by declaring the function**: Make sure the function header properly declares parameters and return type as `double`. This includes defining input types and ensuring the return value can handle the precision needed for fractional numbers. ```c double power(int x, int n) ``` - **Handle Edge Cases**: Write precondition checks, such as ensuring the base is not zero when a negative exponent is provided, to avoid division by zero. - **Initialize a Result Variable**: Start with a variable set to 1.0, because any number to the power of 0 is 1. ```c double result = 1.0; ``` - **Process Positive Exponents**: Multiply the base by itself the exponent number of times using a simple loop. - **Adjust Code for Negative Exponents**: - Convert the exponent to positive. - Multiply base by itself using a loop. - Compute the reciprocal by dividing 1 by the resulting product. - **Return the Result**: Once calculations are complete, returning the result concludes the function process, ensuring accuracy and proper handling of all potential inputs, including edge cases.