`root(100)(100)` Use Newton's method to approximate the given number correct to eight decimal places.

Expert Answers
lfryerda eNotes educator| Certified Educator

Newton's method for the equation `f(x)=0`  is given by `x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}`  where `x_n`  is the nth iteration.  In this case, we need to consider the number `(100)^{1/100}` as equivalent to solving the equation `x^{100}-100=0` which means that `f(x)=x^{100}-100` and `f'(x)=100x^{99}`.

It is now necessary to set up an iteration, and use an initial guess for the answer.

`x_{n+1}=x_n-\frac{x^{100}-100}{100x^99}`      using the Newton's method formula

We can take any reasonable guess for the starting point, so we select `x_1=1.1`.  This gives the iterations:










This last value (1.047128548) is correct to 8 decimal places.