Most real number arithmetic is pretty vacuous
[tex]\sqrt 2 + \sqrt 3 = \sqrt{2} + \sqrt{3}[/tex]
That's about as good a way as possible to write this particular real number. But it's far from the only way.
[tex] (\sqrt 2 + \sqrt 3)^2 = \sqrt{2}^2 + 2\sqrt 2 \sqrt 3 + \sqrt{3}^2 = 5 + 2 \sqrt 6[/tex]
so
[tex]\sqrt 2 + \sqrt 3 = \sqrt{5 + 2 \sqrt 6}[/tex]
Sometimes you can factor something out and you have a common radical:
[tex]\sqrt{32} + \sqrt{128} = \sqrt{2^5} + \sqrt{2^7} = 4 \sqrt{2} + 8 \sqrt{2} = 12 \sqrt{2}[/tex]
But most of them are vacuous,
[tex]3\sqrt{7} + 34 \sqrt{13} = 3\sqrt{7} + 34 \sqrt{13}[/tex]
