I hope you're keeping safe and well. I stumbled across this problem and wondered whether you could help.
Show that $\left(a+\sqrt b\right)\left(a-\sqrt b\right)^3$ is irrational if $a$ and $b$ are NOT square numbers.
Thank you so much for your help in advance. Pac-Man
Let's develop the expression: $\left(a+\sqrt b\right)\left(a-\sqrt b\right)^3=\left(a+\sqrt b\right)\left(a-\sqrt b\right)·\left(a-\sqrt b\right)^2=(a^2-b)(a^2-2a\sqrt b+b)$ We can see that the left term is rational if (but not only if) $a$ and $b$ are rational. On the other hand, the right term is rational if (again, not only if) "$-2a\sqrt b$" is rational. That holds if b is a square. As you are supposing that neither $a$ nor $b$ are squares, this term is irrational, what implies that all the expression is irrational. HOWEVER, that only holds if we suppose that both $a$ and $b$ are rational numbers.
Hope it was useful :)