Question

How do I add and subtract radicals with no like terms

Answer 1

Most real number arithmetic is pretty vacuous

$\sqrt 2 + \sqrt 3 = \sqrt{2} + \sqrt{3}$

That's about as good a way as possible to write this particular real number. But it's far from the only way.

$(\sqrt 2 + \sqrt 3)^2 = \sqrt{2}^2 + 2\sqrt 2 \sqrt 3 + \sqrt{3}^2 = 5 + 2 \sqrt 6$

so

$\sqrt 2 + \sqrt 3 = \sqrt{5 + 2 \sqrt 6}$

Sometimes you can factor something out and you have a common radical:

$\sqrt{32} + \sqrt{128} = \sqrt{2^5} + \sqrt{2^7} = 4 \sqrt{2} + 8 \sqrt{2} = 12 \sqrt{2}$

But most of them are vacuous,

$3\sqrt{7} + 34 \sqrt{13} = 3\sqrt{7} + 34 \sqrt{13}$