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Is there a method by which we can prove that $$\sqrt{3}+\sqrt{7}$$ is irrational. It's obviously an irrational number, but I want to prove that mathematically.

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marked as duplicate by lab bhattacharjee, M Turgeon, Grigory M, Moishe Kohan, Asaf Karagila May 10 '14 at 14:16

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    $\begingroup$ Hint: If it is rational, then so is its square, $\endgroup$ – Geoff Robinson May 10 '14 at 7:54
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    $\begingroup$ I'm just wondering, is the title suggesting there is a way argue non-mathematically about this? $\endgroup$ – Ittay Weiss May 10 '14 at 7:57
  • $\begingroup$ Yes , the title is not the best one indeed @IttayWeiss $\endgroup$ – I.Gandakov May 10 '14 at 8:01
  • $\begingroup$ It's not 100% a duplicate, just similar to some already posted on this site as I see now. $\endgroup$ – I.Gandakov May 10 '14 at 13:21
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Notice that $(\sqrt 3+\sqrt 7)^2=10+2\sqrt{21}$, so that $((\sqrt 3+\sqrt 7)^2-10)^2=84$. This means that the number $\alpha=\sqrt3+\sqrt7$ is a root of the polynomial $$f(X)=(X^2-10)^2-84=x^4-20 x^2+16.$$ You can now use the rational root test to show that $f$ does not have any rational roots: indeed, it follows from that result that all rational solutions are actually integers which divide $16$, and you can easily check that no integer dividing $16$ is a root of $f$.

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Consider \begin{align} &a = \sqrt{7}+\sqrt{3}&&(1)\\ &a(\sqrt{7}-\sqrt{3})=(\sqrt{7}+\sqrt{3})(\sqrt{7}-\sqrt{3})&&(2)\\ &\sqrt{7}-\sqrt{3}=\frac{4}{a}&&(3)\\ &2\sqrt{7}=a+\frac{4}{a}&&\text{summing the reverse of (1) with (3)} \end{align} If $a$ is rational, then also $\sqrt{7}$ is rational.

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If $a:= \sqrt{3} + \sqrt{7}$ were rational, so would $\frac{a^2 - 10}{2}$ be, which equals $\sqrt{21}$. So suppose $\frac{p}{q} = \sqrt{21}$, then $p^2 = 21q^2$. The right hand side has an odd number of prime factors $3$ (a square always has an even number (including $0$) of any prime), and the left hand side has an even number. This is a contradiction.

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