Final answer:
Newton's method involves reformulating the equation into f(x)=0, deriving the function, and then iteratively applying an initial guess and a scheme until convergence to six decimal places.
Step-by-step explanation:
To use Newtons method to find the roots of the equation 8/x = 1 + x3, you must first reformulate it into a form suitable for the method. This requires having a function f(x) equals zero. Let's define f(x) as f(x) = 8/x - 1 - x3. Now, Newton's method makes use of the derivative of the function. The derivative, f'(x), would be derived from f(x) using basic calculus. After obtaining f'(x), you would pick an initial guess x0 and then use the iterative scheme:
xn+1 = xn - f(xn)/f'(xn)
Repeat this process until successive values of xn converge to the desired precision, which in this case is six decimal places. Calculators or computer software can greatly aid in performing these calculations accurately.