Question 1012052
This is the way it plays in my mind.
I would probably not write all the steps I show. {{{3/root(4,144)=3/root(4,12*12)=3/root(4,2*2*3*2*2*3)=3/root(4,2^4*3^2)=3/(root(4,2^4)*root(4,3^2))=3/2root(4,3^2)=root(4,3^4)/2root(4,3^2)=(1/2)root(4,3^4/3^2)=(1/2)root(4,3^2)=(1/2)sqrt(3)=sqrt(3)/2}}}
 
I also like to think of roots as rational exponents,
{{{3/root(4,144)}}}={{{3/144^"1 / 4"}}}={{{3/(12^2)^"1 / 4"}}}={{{3/(12^"2 ( 1 / 4 )")}}}={{{3/12^"1 / 2"}}}={{{
3/(4*3)^"1 / 2"}}}={{{3/(4^"1 / 2"*3^"1 / 2")}}}={{{3/(2*3^"1 / 2")}}}={{{(1/2)(3/3^"1 / 2")}}}={{{(1/2)3^"1 - 1 / 2"}}}={{{(1/2)3^"1 / 2"}}}={{{3^"1 / 2"/2}}}={{{sqrt(3)/2}}}