Why is the square root of a sum not equal to the square root of each its addends? Example: Let's presume one was attempting to isolate m below:

A common mistake would be:
$k^2 = m^2 + n^2 \to k = m +n$

Even though: $k^2 = m^2 + n^2 \to k \neq m +n$

If you apply a square root to both sides of the equation, you will have an inequality.

Why is this true?
If you put the question in a broader context you may understand it better.

Suppose $f$ is some function you can apply to numbers. It might be squaring, or square-rooting, or inverting, or raising to some other power, or taking logarithms or $\sin$ or $\cos$ or just adding 15. In none of these cases is $f(x+y)$ the same as $f(x) + f(y)$ (you should check). In general, you would not expect that coincidence. 

The special case in which it is true is the function "multiply by a fixed quantity". That's the distributive law:

$$
c \times (x + y ) = c \times x + c \times y .
$$

Many of the most common errors students make in algebra or precalculus come from thinking that those other functions behave this way too.

Edit: Just in case you missed @Rahul 's comment: What we need here is a cure for the “law of universal linearity”: Pedagogy: How to cure students of the &quot;law of universal linearity&quot;?
Otherwise
$\sqrt n = n$ for all natural number $n$.

More generally 
$\sqrt q = q$ for all positive rational number $q$.

Homework:

Is it true that:
$\sqrt x  = x$ for all positive real number $x$?