# Does the word 'root' in the context of 'function roots' and 'nth roots' have any relation?

This has been something that in all of my years of mathematics, I haven't ever questioned. Is there any connection to the word 'root' in the context of function roots (e.g. polynomial roots) and nth roots?

In the context of taking the nth root of a number, it's essentially asking 'what number can be raised by n to equal x?'. Thus the 2 root of 32 is asking, 'x^2 = 32: solve for x'

Now in the context of function roots, it's asking 'for what value of the independent variable does the function equal zero?'.

So my question: What's the significance of the word 'root'? What's the history of this word? Why's it being applied to two seemingly different things?

Thanks,

• $f(x) = x^k - a = 0$? – user251257 Nov 22 '16 at 23:49
• @user251257 For some reason I find this really interesting. Is this really the connection? The 'nth root' problem is a just a typical function root problem but with constraints? – Izzo Nov 22 '16 at 23:52
• what constraints? – user251257 Nov 22 '16 at 23:54
• @Teague yes. The n'th root is a simplification of the general finding a polynomial root. – Q the Platypus Nov 22 '16 at 23:55
• The constraint that f(x) = x^k - a – Izzo Nov 22 '16 at 23:55

Any formula that asks you to find the n'th root of a number such as $\sqrt[k]{a}$ can be transformed into an equivalent polynomial root problem $f(x) = x^k - a = 0$. So finding the n'th root is a special case of the general polynomial root.