What is a radical of 136?

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Answer 1 · 2016-09-14T14:43:15

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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:

$(root(n)(136))^n = 136$