# What does the small number on top of the square root symbol mean?

I just came across this annotation in my school's maths compendium:

The compendium is very brief and doesn't explain what this means.

This is the inverse function of $$a^n$$. Hence $$\sqrt[n]a$$ means, you look for a number $$b$$, which when multiplied $$n$$ times with itself results in $$a$$.

For instance: We know that $$2^3 = 8$$, so $$\sqrt[3]8 = 2$$, $$\sqrt[5]{-1}=-1$$ because $$(-1)^5 = -1$$. $$\sqrt[4]3 \approx 1.31607$$ because $$1.31607^4 \approx 3$$.

If there is no number at the top of the root symbol, it means $$n=2$$, so $$\sqrt[2]a = \sqrt a$$.

• Very well explained, thank you – Hubro Aug 13 '13 at 9:41
• I would not recommend to use this notation with negative numbers as you did. – Tom-Tom Sep 23 '15 at 11:48
• Could you explain why $(-1)^5=-1\sqrt[4]{3}$? I'm really confused on that. – Travis Apr 6 '17 at 11:24
• @Travis It is not. These are two examples. The first example is: $(-1)^5 = -1$ and the second one is $\sqrt[4]3 \approx 1.31607$. – Jakube Apr 6 '17 at 12:17
• Oh, didn't see the comma. Thanks. – Travis Apr 6 '17 at 12:19

It means that instead of the "square root of a" you are now considering the "nth root of a". This is the same as writing $a^{1 \over n}$. And just like the square root is "undone" by applying a squared term, i.e., $(\sqrt a)^2 = a$, so the nth root is "undone" by applying the nth power, i.e., $(\sqrt[n] a)^n = a$.

$\sqrt[n] a$=$a^ \frac 1 n$

Also,if $\sqrt[n] a$=$x$ then $x^n=a$

• @Olorun-Didn't notice that it was 2 year old...may be it was recently modified....saw it in the newest questions tab... – tatan Sep 23 '15 at 6:29
• Different answers teach different and sometimes new things, and I see no problem with new answers, given to old posts. – codezombie Dec 26 '16 at 11:20
• Minor point: I notice quite a few elementary algebra books as well as some writers here taking the view that the n-th root of x is defined as x to the power 1/n. I disagree strongly. For an elementary student, the idea of saying "what raised to the n-th power gives x?" is more straightforward than saying "just raise x to the power 1/n" particularly since the way one does this without a calculator is the former, not the latter. That is, x to the power 1/n should be the entity being defined, with it defined as I've described - and not the reverse. – Dr. Michael W. Ecker Dec 11 '19 at 7:16