Posted by Paula on Friday, July 26, 2013 at 12:44am.
There is no counterexample. The fifth root is indeed sometimes irrational.
Pick any integer which is not a perfect 5th power, and its 5th root is irrational.
5th powers: 0,1,32,343,1024,3125,...
These all have rational 5th roots: 0,1,2,3,4,5,...
Any numbers in between those have irrational 5th roots.
Not only is the 5th root sometimes irrational, it is "almost always" irrational. And yet, infinity is a strange animal. While it seems like the vast majority of integers ought to have irrational 5th roots, there are just as many which have rational 5th roots.
find the value of 7+2(5-2.3^2).
why? Have you no calculator?