Such A Long sequence

Algebra Level 4

131+3+153+35...........+1225223+223225=?\large \frac{1}{3\sqrt1+\sqrt3}+\frac{1}{5\sqrt3+3\sqrt5}...........+\frac{1}{225\sqrt{223}+223\sqrt{225}} = ?

If this sum can be represented as ab\frac{a}{b} , where both a,ba,b are square free and positive integers, find (a$b)\large(a\$b) if the operation (a$b)\large(a\$b) is defined as b(moda) \large b \pmod{a}.

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