prove that there are real numbers a and b such that square root of a + square root of b equals square root of a + b

Lets firts mention that the rule presented in the question does not apply for all real numbers

But, we can see that in some cases it might be true.

Let

a = 1

b = 0

√1 + √0 = 1 + 0 = 1

Likewise,

√ (1 + 0) = √ (1) = 1

And we see that the result is the same.