Suppose we try to find a 'simpler' expression than #sqrt(a^2+b^2)#

Such an expression would have to involve square roots or #n#th 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.