See explanation…
The first kind of radical you meet is a square root, written:
##sqrt(136)##
This is the positive irrational number (##~~11.6619##) which when squared (i.e. multiplied by itself) gives ##136##.
That is:
##sqrt(136) * sqrt(136) = 136##
The prime factorisation of ##136## is:
##136 = 2^3*17##
Since this contains a square factor, we find:
##136 = sqrt(2^2*34) = sqrt(2^2)*sqrt(34) = 2sqrt(34)##
Note that ##136## has another square root, which is ##-sqrt(136)##, since:
##(-sqrt(136))^2 = (sqrt(136))^2 = 136##
Beyond square roots, the next is the cube root – the number which when cubed gives the radicand.
##root(3)(136) = root(3)(2^3*17) = root(3)(2^3)root(3)(17) = 2root(3)(17) ~~ 5.142563##
For any positive integer ##n## there is a corresponding ##n##th root, written:
##root(n)(136)##
with the property that:
##(root(n)(136))^n = 136##
