You have such entry data:
[tex]1\ \textless \ a\ \textless \ 10 \\ 1\ \textless \ b\ \textless \ 36[/tex]
Consider expression [tex] \sqrt{a+ \sqrt{b} } [/tex]. If this expression becomes an integer, then b=4,9,16,25, because then [tex] \sqrt{b} =[/tex] 2,3,4,5, respectively. In other cases [tex] \sqrt{b} [/tex] is not integer and thus the expression [tex] \sqrt{a+ \sqrt{b} } [/tex] also is not integer.
1. b=4, then [tex] \sqrt{a+ \sqrt{b} }=\sqrt{a+2} [/tex]. Here a=2,7 (in other cases [tex] \sqrt{a+2} [/tex] is not integer). When a=2, [tex] \sqrt{a+2}=\sqrt{2+2}=2 [/tex] and when a=7, [tex] \sqrt{a+2}=\sqrt{7+2}=3 [/tex].
2. b=9, then a=6 and [tex]\sqrt{a+ \sqrt{b} }= \sqrt{a+3}=\sqrt{6+3}=3 [/tex].
3. b=16, then a=5 and [tex]\sqrt{a+ \sqrt{b} }= \sqrt{a+4}=\sqrt{5+4}=3 [/tex].
4. b=25, then a=4 and [tex] \sqrt{a+ \sqrt{b} }=\sqrt{a+5}=\sqrt{4+5}=3 [/tex].
Answer: (2,4), (7,4), (6,9), (5,16), (4,25).
