# What is a radical of 136?

Sep 14, 2016

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#### Explanation:

The first kind of radical you meet is a square root, written:

$\sqrt{136}$

This is the positive irrational number ($\approx 11.6619$) which when squared (i.e. multiplied by itself) gives $136$.

That is:

$\sqrt{136} \cdot \sqrt{136} = 136$

The prime factorisation of $136$ is:

$136 = {2}^{3} \cdot 17$

Since this contains a square factor, we find:

$136 = \sqrt{{2}^{2} \cdot 34} = \sqrt{{2}^{2}} \cdot \sqrt{34} = 2 \sqrt{34}$

Note that $136$ has another square root, which is $- \sqrt{136}$, since:

${\left(- \sqrt{136}\right)}^{2} = {\left(\sqrt{136}\right)}^{2} = 136$

Beyond square roots, the next is the cube root - the number which when cubed gives the radicand.

$\sqrt[3]{136} = \sqrt[3]{{2}^{3} \cdot 17} = \sqrt[3]{{2}^{3}} \sqrt[3]{17} = 2 \sqrt[3]{17} \approx 5.142563$

For any positive integer $n$ there is a corresponding $n$th root, written:

$\sqrt[n]{136}$

with the property that:

${\left(\sqrt[n]{136}\right)}^{n} = 136$