Prove that for any distinct primes $p$ and $q$, the ratio $\frac{\sqrt p}{\sqrt q}$ is irrational.
I know that separately $\sqrt p$ and $\sqrt q$ are irrational, so my initial thought process was to show that they are each irrational, but it is not always true that an irrational number divided by an irrational number is also irrational, could someone lead me in the right direction?
We show it using a proof by contradiction.
Suppose that you can write $\sqrt{\frac{p}{q}}$ = $\frac{a}{b}$ Where p and q are primes and a,b are rational numbers and that $\frac{a}{b}$ cannot be reduced.
Then $\frac{p}{q} = \frac{a^2}{b^2}$
(1) $\frac{p}{q} *{b^2}= {a^2}$
Then a is a multiple of $\frac{p}{q}$.
(2) $a = k * \frac{p}{q}$
Insert (2) in (1): $\frac{p}{q} *{b^2}= {(k * \frac{p}{q})^2}$
$\frac{p}{q} *{b^2}= {k^2 * \frac{p^2}{q^2}}$
Divide by $\frac{p}{q}$:
${b^2}= {k^2 * \frac{p}{q}}$
We see that both b and a have common factors, but due to our initial statement that a and b have no common factors, $\sqrt{\frac{p}{q}}$ is not rrational.