Here's what I've thought so far. Unless I'm very much mistaken, we have that $[\mathbb{Q}(\sqrt{3},\sqrt{5}):\mathbb{Q}] = 15$, so I'm looking for a $\theta$ such that $[\mathbb{Q}(\theta):\mathbb{Q}] = 15$. Because it's supposed to be a simple extension, I need the minimal polynomial of $\theta$ over $\mathbb{Q}$ to be of degree $15$. I'm quite sure $\theta$ cannot be $15$, although I have only briefly sketched out why I think that is.
I'm looking for some hints on how to approach this problem. I'll update this post with an edit once I have "solved" it and hopefully someone will tell me if it's correct or not.