Explain why sqrt(a+b) does not equal sqrt a + sqrt b, for a>0 and b>0?

Answer 1

because

(a+b)^2 ≠ a^2 + b^2

sqrt is not a linear operator

Answer 2

How about an indirect proof:

Assume that √(a+b) = √a + √b , for all a, b > 0

all we need is one counter-example to show my assumption is false:

is √(9 + 16) = √9 + √16 ???
LS = √25 = 5
RS = 3 + 4 = 7
RS ≠ LS

so I have a case where my assumption is false, all done