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.

Related questions
See all questions in Simplification of Radical Expressions
Impact of this question
1629 views around the world
You can reuse this answer
Creative Commons License