Yes, it's possible.
Suppose [tex]\sqrt a+\sqrt b[/tex] and [tex]\sqrt{a+b}[/tex] are equal. Take squares on both sides.
[tex]\left(\sqrt a + \sqrt b\right)^2 = \left(\sqrt{a+b}\right)^2[/tex]
[tex]\left(\sqrt a\right)^2 + 2\sqrt a\sqrt b + \left(\sqrt b\right)^2 = a + b[/tex]
[tex]a + 2\sqrt a \sqrt + b = a + b[/tex]
[tex]2\sqrt a \sqrt b = 0[/tex]
This can only be true if either [tex]\sqrt a=0[/tex] or [tex]\sqrt b=0[/tex], or equivalently if [tex]a=0[/tex] or [tex]b=0[/tex].
Now choose [tex]a=0[/tex] and [tex]b[/tex] any other positive number. Then
[tex]\sqrt 0 + \sqrt b = \sqrt b[/tex]
and
[tex]\sqrt{0+b} = \sqrt b[/tex]
are clearly the same.
