Is the $\sqrt{(-10)^2}$ equivalent to $-10$ or $10$?

Question

Is $\sqrt{(-10)^2}$ equivalent to $-10$ or $10$, or is it equivalent to only one among the two?

Since $\sqrt{(-10)^2} = \sqrt{100}$ and $\sqrt{100} =$ $-10$ or $10$. Using this solution, it can be equivalent to either the two answers.

But using this solution:

$\sqrt{(-10)^2} = -10^{\frac{2}{2}}$

$-10^{\frac{2}{2}} = -10$

It has only 1 answer.

Answer 1

We know that the function $y=x^2$ isn't invertible in all the domain, so in order to find the inversr, we can consider: $$f^{-1}: R^+\rightarrow R^+$$ So, the correct answer is $\sqrt{10^2}=10$ and you can check this using a graph calculator.

Note that evn if you are using complex numbers, then you obtain again $\sqrt{(-10)^2}=10$ and not $\pm10i$ because: $$(\pm10i)^2=-100$$

Answer 2

$$\sqrt x$$ is a function of $x$. As such, it can only take one value per $x$.