Why can't the square root of a^2 + b^2 be simplified?

Question

Julia Adams · Accepted Answer

If we substitute a and b to equal 6 for example
it would be  #sqrt(6^2+6^2)# it would equal 8.5(1.d.p) as it would be written as #sqrt(36+36)# giving a standard form as #sqrt72# However if it was #sqrt6^2+sqrt6^2# it would equal 12 as the #sqrt# and #^2# would cancel out to give the equation 6+6 Therefore #sqrt(a^2+b^2)# cannot be simplified unless given a substitution for a and b. I hope this isn't too confusing.

Eva Adamson · Answer

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.