What is the cube root of #47.1# ?

1 Answer
George C.
Mar 21, 2017

#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.

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