An algebra problem by أحمد الحلاق

Algebra Level 4

\[ \large \left(1 + \dfrac1{1 + 2^1} \right)\left(1 + \dfrac1{1 + 2^2} \right)\left(1 + \dfrac1{1 + 2^3} \right)\cdots \left(1 + \dfrac1{1 + 2^{10}} \right) \]

If the expression above can be expressed as \( \dfrac mn\), where \(m\) and \(n\) are coprime positive integers, find \(m+n\).