Question 326842
Let's find out.
Square both sides.
{{{sqrt(a)+sqrt(b)=sqrt(a+b)}}}
{{{(sqrt(a)+sqrt(b))^2=a+b}}}
{{{a+2sqrt(ab)+b=a+b}}}
{{{2sqrt(ab)=0}}}
{{{sqrt(ab)=0}}}
Yes, it's true but only when {{{a=0}}} or {{{b=0}}} or both. 
So in general, other than those special cases, it does not hold.