Suppose we try to find a 'simpler' expression than sqrt(a^2+b^2)
Such an expression would have to involve square roots or nth roots or fractional exponents somewhere along the way.
Hayden's example of sqrt(6^2+6^2) shows this, but let's go simpler:
If a=1 and b=1 then sqrt(a^2+b^2) = sqrt(2)
sqrt(2) is irrational. (Easy, but slightly lengthy to prove, so I won't here)
So if putting a and b into our simpler expression only involved addition, subtraction, multiplication and/or division of terms with rational coefficients then we would not be able to produce sqrt(2).
Therefore any expression for sqrt(a^2+b^2) must involve something beyond addition, subtraction, multiplication and/or division of terms with rational coefficients. In my book that would be no simpler than the original expression.