# What is the remainder of an n-th root called?

I feel like there should be a better word than remainder, but I don't know it. What do you call the thing that's left over when performing an $n$-th root? For example, $\sqrt[3]{29}$ is $3$ with 2 left over, while $\sqrt[3]{63}$ is also 3, but with $36$ left over. Is there a preferred notation for this quantity?

• What would be your so-called 3-rd root remainder of $-15$? I.e. how do you want to define it for negative reals with odd roots? – user26486 Jun 17 '15 at 18:58
• @user26486 Great question. The particular problem I'm working on doesn't have any negative arguments. Without knowing how the function would be used for negative numbers, I'm at a loss as to whether that'd be $\pm 7$. For complex answers (which I wouldn't see in this problem) I also have no idea about what a 'right' answer would be. – user121330 Jun 17 '15 at 19:57

Let your wanted function be $r_n(a)$. If $a\ge 0$, then (in your case $n=3,a\in\{29,63\}$):
$$r_n(a)=a-\lfloor \sqrt[n]{a}\rfloor^n$$
Here I've used the floor function: $\lfloor x\rfloor$ is defined as the largest integer less than or equal to $x$.
• Is $r(n)$ a remainder? Is there another word for it? It appears you're saying that there is no standard notation for whatever this quantity is called, is that accurate? – user121330 Jun 17 '15 at 19:59
• @user121330 You should understand what my created notation $r_n(a)$ means by looking at the example I gave (namely $n=3, a\in\{29,63\}$, which outputs your given remainders). It is a function that gives your wanted remainder. I believe there is probably no standard notation for it. – user26486 Jun 17 '15 at 20:06