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##