Suppose you didn’t know that 3√64 = 4. Write down Newton’s method for solving f(x) =x^3 − 64 = 0. Choose x0 = 5 as your initial guess for the solution of this equation and iterate until the first 5 significant figures of your solution are correct.

Newton's method is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The method starts with a function f defined over the real numbers x, an initial guess x0 for a root of f, and the function's derivative f'.

The formula for Newton's method is:

x_(n+1) = x_n - f(x_n) / f'(x_n)

Given the function f(x) = x^3 - 64, the derivative f'(x) = 3x^2.

Let's start with the initial guess x0 = 5.

1. First iteration: x1 = x0 - f(x0) / f'(x0) = 5 - (5^3 - 64) / (3 * 5^2) = 5 - (125 - 64) / 75 = 5 - 61 / 75 = 4.18666666667

2. Second iteration: x2 = x1 - f(x1) / f'(x1) = 4.18666666667 - ((4.18666666667)^3 - 64) / (3 * (4.18666666667)^2) = 4.00225733634

3. Third iteration: x3 = x2 - f(x2) / f'(x2) = 4.00225733634 - ((4.00225733634)^3 - 64) / (3 * (4.00225733634)^2) = 4.00000063669

4. Fourth iteration: x4 = x3 - f(x3) / f'(x3) = 4.00000063669 - ((4.00000063669)^3 - 64) / (3 * (4.00000063669)^2) = 4.00000000000

After the fourth iteration, the first five significant figures are correct. Therefore, the root of the equation x^3 - 64 = 0 is approximately 4.00000.