What is a radical of 136?

1 Answer
Sep 14, 2016

Answer:

See explanation...

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#