Such A Long sequence

Algebra Level 4

\(\large \frac{1}{3\sqrt1+\sqrt3}+\frac{1}{5\sqrt3+3\sqrt5}...........+\frac{1}{225\sqrt{223}+223\sqrt{225}} = ?\)

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

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