Would the answer of the square root of a square root be positive or negative?

Question

I have this problem:

$$\sqrt{\sqrt{16}}$$ would it be positive 2 or it would be $\pm$2?

In this class we do not deal with complex numbers, so all the square roots are positive, thus for the class the answer is 2.

However, I am asking this question: in real mathematics, doesn't one consider the negative root as well?

Answer 1

The symbol $\sqrt{x}$ always denotes the non-negative square root of $x$, assuming that $x\ge 0$. Thus,

$$\sqrt{\sqrt{16}}=\sqrt4=2\;.$$

Answer 2

As Brian said by $\sqrt{x}$ we usually (I wouldn't say always) mean the non-negative square root of x. But because this is for class needs and you'll usually meet this in solving equation you should always consider both negative and positive root, because they can lead to two distinct solution.

Why we usually mean only the non-negative square is because in some fields (e.g. geometry) there aren't negative values. So if you take square root of given area of a square, you'll get the side of the square, but the negative value is left out, because square with negative sides doesn't exist.

So it depends on where you use it. In geometry we use only the positive root, but in algebra we must consider both roots.

Answer 3

Equivalently, we know that:

$$\sqrt{\sqrt{16}}=\sqrt[4]{16}=\sqrt[4]{(\pm 2)^4}=|\pm2|=2$$

Answer 4

for the solution to $x^4=16$ the answers are $x\in \{2,-2,2i,-2i\}$

this is because you need to combine the answers of $x^2=4$ and $x^2=-4$