1. What you have to do is to find the minimal polynomial of [tex]\sqrt{7}[/tex] and [tex]\sqrt{5}[/tex], that wil give you the degree [tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}][/tex].
Remark, note that
[tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{5},\sqrt{7}),\mathbb{Q}(\sqrt{7})][\mathbb{Q}(\sqrt{7}):\mathbb{Q}][/tex]