Proving that $(\sqrt 3+\sqrt 2)/(\sqrt 2)$ is irrational

Question

Can you please help me prove this ? I can prove that $\sqrt p$, where p is prime is irrational, also that the sum $\sqrt 3 + \sqrt 2$ is irational, bud dont know how to prove that the whole fraction is irrational. Thanks for any answer.

Answer 1

$$\frac{\sqrt3+\sqrt2}{\sqrt2}=1+\sqrt{3/2}$$ The whole fraction is irrational iff $\sqrt{3/2}=\frac12\sqrt6$ is irrational iff $\sqrt6$ is irrational. The last statement you should be able to prove.

Answer 2

If $\frac{\sqrt3+\sqrt2}{\sqrt2}\in\mathbb Q$ so $\sqrt{\frac{3}{2}}\in\mathbb Q.$

Now, let $\sqrt{\frac{3}{2}}=\frac{m}{n},$ where $m$ and $n$ are naturals such that $\gcd(m,n)=1$, and get a contradiction.

Can you end it now?

Answer 3

Consider $r=(\sqrt{3}+\sqrt{2})/\sqrt{2}$. Then $$ 3=2(r-1)^2 $$ The possible positive rational roots of $$ 2x^2-4x+1 $$ are $1$ and $1/2$, neither of which is actually a root.