Question

What is a radical of 136?

Answer 1

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`