Irrationality of $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$ Sorry, for a rather silly question.
Suppose $a_1$, $b_1$, $a_2$, $b_2$ are integers, all different from zero, while $b_1$ and $b_2$ are co-prime positive integers, neither being a complete square.
Is there an elementary proof that $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$ is irrational? Or maybe it's just wrong?
Multiplying gives three distinct surds $n_1\sqrt{b_1}$, $n_2\sqrt{b_2}$, $n_3\sqrt{b_1\;b_2}$, so it doesn't seem to help, while taking a power gives again a product of two irrationals.
How about more general case $(a_1+\sqrt[m_1]{b_1})(a_2+\sqrt[m_2]{b_2})$ ?

$\sqrt a = r + s\sqrt b\ $ follows from $\, c + \sqrt{a} = \dfrac{e}{d+\sqrt b}\,$ by rationalizing  denominator. Now apply
Theorem $ $ If $\,r,s\in\Bbb Q\,$ and $\,\sqrt a = r + s\sqrt b\,$ then $\sqrt a,\sqrt b\not\in \Bbb Q\Rightarrow \sqrt{ab}\in\Bbb Q$
Proof $\, $ Squaring $\Rightarrow a = r^2\!+s^2b + 2rs\, \sqrt b\ $ so $\ rs = 0\,$ (else solving for $\sqrt b\,$ $\,\Rightarrow \sqrt b\in \Bbb Q)$.
$s\neq 0\,$ (else $\,a = r^2,\,$ contra $\,\sqrt a \not\in \Bbb Q)\,$ so $\,r=0\,$ so $\,\sqrt a = s\sqrt b\,$ $\overset{\large\times\sqrt b}\Longrightarrow\sqrt{ab} = sb\in \Bbb Q$.
Note $ $ The proof can be reformulated in field theoretic language as the Lemma below (which inductively generalizes to any number of sqrts and also $n$'th roots).

Lemma $\rm\ \ [K(\sqrt{a},\sqrt{b}) : K] = 4\ $ if  $\rm\ \sqrt{a},\ \sqrt{b},\ \sqrt{a\,b}\ $  all are not in $\rm\,K\,$ and $\rm\, 2 \ne 0\,$ in the field $\rm\,K.$
Proof $\  $  Let  $\rm\ L = K(\sqrt{b}).\,$ Then $\rm\,  [L:K] = 2\,$  via  $\rm\,\sqrt{b}  \not\in K,\,$  so it suffices to prove $\rm\, [L(\sqrt{a}):L] = 2.\,$ It fails only if  $\rm\,\sqrt{a} \in L = K(\sqrt{b})\, $ and then $\rm\, \sqrt{a}\ =\  r + s\, \sqrt{b}\ $  for $\rm\ r,s\in K.\,$ But that's impossible,
since squaring  $\Rightarrow \rm(1)\!:\ \ a\ =\ r^2 + b\ s^2 + 2\,r\,s\  \sqrt{b},\, $ which contradicts hypotheses as follows:
$\rm\qquad\qquad rs \ne 0\ \ \Rightarrow\ \  \sqrt{b}\ \in\  K\ \ $ by solving $(1)$ for $\rm\sqrt{b}\,,\,$ using  $\rm\,2 \ne 0$
$\rm\qquad\qquad\  s = 0\ \ \Rightarrow\ \  \ \sqrt{a}\ \in\  K\ \ $  via  $\rm\ \sqrt{a}\ =\ r+s\,\sqrt b = r \in K$
$\rm\qquad\qquad\  r = 0\ \ \Rightarrow\ \  \sqrt{a\,b}\in K\ \ $  via  $\rm\ \sqrt{a}\ =\ s\, \sqrt{b},\ \ $times $\rm\,\sqrt{b}\quad$
