Hey sorry I've been working on it so haven't had a chance to check back. Thanks for the help! So far I've written this:
Code:
import java.io.*;

public class Assignment1try2

{

    public static void main(String[] args)

    {
        //Exports the data to an excel file

     PrintWriter outputFile = null;
       try
         {
           outputFile = new PrintWriter(new FileOutputStream("outputVelomobile.xls",false));
          }
            catch(FileNotFoundException e)
         {
           System.out.println("File error.  Program aborted.");
           System.exit(0);
          }


        //Variable declarations for the Quest Velomobile

    double n = 0.92;
    double rho = 1.20;
    double cR = 0.004;
    double m = 32+80;
    double g = 9.81;
    double cD = 0.24;
    double area = 0.46;
    double valueOne = rho*cD*area;
    double valueTwo = cR*m*g;
    double initialRootEstimate = 1;
    double function, absDifference;
    double[] rootEstimate = new double[8];
    int numLoops = 0;
    int[] powerRider = {25,50,75,100,150,200,300,400};


        //Start loop
    for (int index=0;index<powerRider.length;index++)

    {

         numLoops = 0;

        do
        {

        function = -n*powerRider[index] + 0.5*(valueOne*(Math.pow(initialRootEstimate,3))) + (valueTwo*initialRootEstimate);

        rootEstimate[index] = initialRootEstimate - (function/((3/2)*(valueOne*(Math.pow(initialRootEstimate,2))) + (valueTwo)));

        absDifference = Math.abs(rootEstimate[index] - initialRootEstimate);

        initialRootEstimate = rootEstimate[index];
        numLoops++;

         }

           while (absDifference >=1.0E-12);

            //Displays data on screen
            System.out.println();
            System.out.println("The Power is:"    +   powerRider[index]);
            System.out.println("The Root Estimate is:"    +   initialRootEstimate);
            System.out.println("The number of cycles is:"    +   numLoops);
            System.out.println();

            //Outputs results on screen

            outputFile.println();
            outputFile.println("The Power is:"    +   powerRider[index]);
            outputFile.println("The Root Estimate is:"    +   initialRootEstimate);
            outputFile.println("The number of cycles is:"    +   numLoops);
            outputFile.println();


        }
            outputFile.close();
    }

}
It works but the only problem is that we end up with a lot of iterations. Is there any way we can reduce the amount of iterations?

thanks again