What exactly IS a square root?

It’s come to my attention that I don’t actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the answer! Obviously square root can be rewritten as $x^{1/2}$ , but how does one actually multiply something by itself half a time?

How do calculators perform the operation?

Answer

How does one actually multiply something by itself half a time ?

Zen Buddhism has a similar question: What is the sound of one hand clapping ? When my father told me, in passing, one day, that $\sqrt x=x^{1/2}$ I had pretty much the same reaction. But then I started thinking to myself: What is the fundamental property of an n-th root, $\sqrt[\Large^n]x$ ? It is basically the number which, when multiplied n times with itself, yields the desired value. At the same time, $x^{1/n}$ multiplied n times with itself, also yields the same value, since $\big(x^a\big)^b=x^{ab}$.

Attribution
Source : Link , Question Author : nuggetbram , Answer Author : Lucian

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