Question

Explain how the process of combining radicals through addition and subtraction is similar to combining polynomials. What makes two radicals like radicals? Give an example.

Answer 1

With polynomials, the variable represents some unknown quantity. While that quantity remains unknown, we can't add or subtract unlike powers of that unknown quantity. With fully reduced radicals, they represent a quantity that is known, but unable to be reconciled (incommensurable) with the integers or any ratio between integers. So in a weird kind of way, their exact quantity is kind of unknown (hence the non-repeating, endless decimal expansion). As such, adding or subtracting unlike, fully-reduced irrationals isn't possible (because, what "exactly" are you adding together?), so the best we can do is combine like instances of the irrational to end up with some multiple of it.