Originally Posted by

**mfetch22** Is the following number (with the added conditions taken into account) always irrational? And how would one go about proving this? Heres the number and the conditions:

Is the number $\displaystyle \sqrt{a} + \sqrt{b}$ always irrational? Given these conditions:

$\displaystyle a \neq b$

$\displaystyle a = 1,2,3.... \;\;\; and \;\;\;b=1,2,3,....$

$\displaystyle a \neq x^m$ for some numbers $\displaystyle x=1,2,3....$ and $\displaystyle m=1,2,3...$

$\displaystyle b \neq y^n$ for some numbers $\displaystyle y=1,2,3....$ and $\displaystyle n=1,2,3..$

There, I think I've covered all the proper conditions. But if there is one I missed, I'm sure you catch the drift of the exact question I am asking, and can probably guess whatever the missing condition is, not that there is a missing condition, just considering the possibility. Anyway, theres the question. Also, I understand the possibility that there is a simple example that shows the number $\displaystyle \sqrt{a} + \sqrt{b}$ to be rational in some obvious case, and if there is such a case, then this question is solved simply. And if your answer is yes, I"d be interested in seeing a proof, or atleast the "gist" of the proof. Thanks in advance