In Gallian's Abstract Algebra, he uses an example of $\{1, \sqrt{3}\}$ being a basis of $\Bbb Q(\sqrt{3}, \sqrt{5})$ over $\Bbb Q(\sqrt{5})$ and $\{1, \sqrt{5}\}$ being a basis of $\Bbb Q(1, \sqrt{5})$ over $\Bbb Q$.
Why is $\{1, \sqrt{3}\}$ a basis of $\Bbb Q(\sqrt{3}, \sqrt{5})$ over $\Bbb Q(\sqrt{5})$? I don't really understand what $\Bbb Q(\sqrt{3}, \sqrt{5})$ is as a set and how I would form a basis from it. I know that $\Bbb Q(\sqrt{5}) = \{a_0 + a_1 \sqrt{5} : a_i \in \Bbb Q \}$, but for multiple "argument" extension fields I don't understand how they'd be written.
Hint
$$a + b \sqrt{3} + c \sqrt{5} + d \sqrt{3}\sqrt{5}=(a+c \sqrt{5}) + (b+d\sqrt{5}) \sqrt{3}$$