How can a square root be defined since it has two answers?

Take it this way: $a = \sqrt{16}$ is just a symbol denoting something.
It denotes (by definition) the non-negative root of $x^2 = 16$.
Since $16 = a^2 = (-a)^2$, it means the other solution of this equation is $-a = -\sqrt{16}$

Deciding whether $1/0$ is positive or negative is not really a problem. The real problem is there is no number $x$ such that $0 \times x = 1$. In contrast, there are exactly two distinct numbers $x_+$ and $x_-{}$ such that $(x_+)^2 = (x_-)^2 = 16$.

It's useful to be able to say sometimes which of the possible square roots we mean in a certain expression. If you want the negative root, write $-\sqrt{16}$. If you want to leave open the possibility of using either square root, write $\pm\sqrt{16}$ as is done in most presentations of the formula for the roots of a quadratic equation.

In general, if $x$ is a positive number, there are $n$ complex numbers whose $n$th power is $x$. But there is only one real, positive number. That is defined to be $x^{1/n}$.

This has to be taken into account when solving equations.

If you have (amongst other constraints) : $$x^2 = 16$$

then at first take you must consider:

Not:

$$x = \sqrt{16} $$

But:

$$x = \pm \sqrt{16}$$