Question

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

Answer 1

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.

Answer 2

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.