# What is the cube root of #47.1# ?

##### 1 Answer

Mar 21, 2017

#### Explanation:

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

If you have a calculator with

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

Otherwise, we can find rational approximations as follows:

Given an approximation

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

Putting

#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.