The tutor above thought you just wanted an alternate expression for a2+b2.
But you wanted a numerical solution.

By inspection we see that a=1 and b=1, and .
is a solution.  But is that the ONLY possible solution?

We might guess that that is the only solution, but guessing
doesn't count!  :)

So let's see if we can prove that it is the only solution.



Clear of fractions and set up the quadratic in b







The discriminant must be a perfect square for b to
even be rational:



So  must be a perfect square, say p2





The discriminant must also be a perfect square for 'a' to
even be rational:



So  must be a perfect square, say q2







The only possibilities are

q-p=1, q+p=8, but that gives a fraction solution.
q-p=2, q+p=4 which gives the only solution in integers,
q=3, p=1 

So



becomes







So we have proved that 'a' can only be 1.









And now b can only be 1 also.

Therefore the only solution for a and b in positive integers is 
a = b = 1, and 

a2+b2 = 12+12 = 1+1 = 2.

Edwin