# How to prove $a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+…+a_{n}\sqrt[b_{n}]{c_{n}}$ is irrational?

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?

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