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Algebra Level 3

\[\large x=\frac { 4 }{ \left( \sqrt { 5 } +1 \right) \left( \sqrt [ 4 ]{ 5 } +1 \right) \left( \sqrt [ 8 ]{ 5 } +1 \right) \left( \sqrt [ 16 ]{ 5 } +1 \right) } \]

Find \(\left( x+1 \right) ^{ 16 }\).

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