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

2 Answers
May 31, 2015

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.

May 31, 2015

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.