Let \(x={4\over{(\sqrt5+1)(\root 4\of5+1)(\root 8\of5+1)(\root {16}\of5+1)}}.\) Find \((x+1)^{48}\)
+23245 Let's multiply both the numerator and the denominator by 51/16 - 1:
The numerator becomes: 4·(51/16 - 1)
The denominator becomes: (51/2 + 1)(51/4 + 1)(51/8 + 1)·[ (51/16 + 1)(51/16 - 1) ]
= (51/2 + 1)(51/4 + 1)(51/8 + 1)·[ (51/8 - 1) ]
re-group: = (51/2 + 1)(51/4 + 1)·[ (51/8 + 1)(51/8 - 1) ]
= (51/2 + 1)(51/4 + 1)·[ (51/4 - 1) ]
re-group: = (51/2 + 1)·[ (51/4 + 1)(51/4 - 1) ]
= (51/2 + 1)·[ (51/2 - 1) ]
= 5 - 1
= 4
Fraction: 4·(51/16 - 1) / 4 = 51/16 - 1
[ ( 51/16 - 1 ) + 1 ]48 = [ 51/16 ]48 = 53 = 125
