# Approximate the 4th root of 75 using Newtowns Method

Approximate $\sqrt[4]{75}$ using the Newton Raphson method

Newton's method is an iterative technique used to find successively better approximations to the roots of a real-valued function. To approximate the fourth root of 75 using Newton's method, we start by defining the function $f(x) = x^4 - 75$ . Our goal is to find the root of this function, which corresponds to the value of x that makes f(x) = 0. Starting with an initial guess, we apply the iteration formula $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ , where f'(x) is the derivative of f(x). For our function, $f'(x) = 4x^3$ . By repeating this process, each iteration brings us closer to the actual fourth root of 75. This method showcases the power of calculus in solving complex numerical problems through approximation.

## Related Problems

Use the Newton Raphson method to approximate the real zero close to $x = 1$ until two successive approximations differ by less than 0.005 for the following function

$f(x) = 2x^2 - 3$

Starting with an initial value $x_1 = 1$, perform 2 iterations of Newton's Method on $f(x) = x^3 - x - 1$ to approximate the root.

Use Newton's Method to approximate the solution to the following equation

$\cos{(x)} = \frac{x}{5}$

Use Newton's Method to approximate a solution to $2\cos{(x)} = 3x$ (Let $x_0 = \frac{\pi}{6}$ and find $x_2$)