Question

What is the cube root of `47.1` ?

Answer 1

`root(3)(47.1) ~~ 3.6113837181884034`

Explanation:

`root(3)(47.1)` is an irrational number somewhere between `3` and `4`, since:

`3^3 = 27 < 47.1 < 64 = 4^3`

If you have a calculator with `e^x` and `ln x` functions, then you can calculate an approximation using:

`root(3)(47.1) = e^(1/3 ln(47.1)) ~~ 3.61138372`

Otherwise, we can find rational approximations as follows:

Given an approximation `a_0`, iterate using the formula:

`a_(i+1) = a_i+(47.1 - a_i^3)/(3a_i^2)`

Putting `a_0 = 3.5` into a spreadsheet, I got the following sequence of approximations:

`a_0 = 3.5`

`a_1 ~~ 3.6149659863945578`

`a_2 ~~ 3.6113872668819988`

`a_3 ~~ 3.6113837181918909`

`a_4 ~~ 3.6113837181884034`

`a_5 ~~ 3.6113837181884034`

As you can see, this converges quite rapidly.