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Let's define the number

$$A=a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+.....+a_{n}\sqrt[b_{n}]{c_{n}}$$

where $a_{1}, a_{2}, ..., a_{n}$ are positive integers and $b_{1}, b_{2}, ..., b_{n}, c_{1}, c_{2}, ..., c_{n}>1$ are positive integers.

For every $1\leq i\leq n$, $c_{i}$ cannot be divided by any $p^{b_{1}}$ where $p$ is a prime number.

How to prove $A$ is irrational?

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    $\begingroup$ Can you prove it for n=2? $\endgroup$ – marty cohen Feb 15 '16 at 22:40
  • $\begingroup$ It's not true as it's currently stated. Take for example $n=1$, then $1\cdot \sqrt[2]{4}$ is not irrational $\endgroup$ – vrugtehagel Feb 15 '16 at 22:52
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    $\begingroup$ @vrugtehagel : $4$ can be divided by $p^{b_1}$, where $p=2,b_1=2$. $\endgroup$ – Watson Feb 15 '16 at 22:55
  • $\begingroup$ Excuse me, I thought that was a result that the OP got from given conditions. I didn't realize it was a condition. $\endgroup$ – vrugtehagel Feb 15 '16 at 22:56
  • $\begingroup$ Related: math.stackexchange.com/questions/440453 $\endgroup$ – Watson Nov 27 '18 at 15:37

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