On expanding the given expression we get, (4√5+5√4)100=
100
∑
r=0
100Cr(4√5)100−r(5√4)r (4√5+5√4)100=
100
∑
r=0
100Cr5
100−r
4
4
r
5
=
100
∑
r=0
Tr+1 Here, Tr+1=100Cr5
100−r
4
4
r
5
Clearly, Tr+1 will be an integer if
100−r
4
and
r
5
are integers. This is possible when 100−r is an multiple of 4 and r is multiple of 5. When 100−r is a multiple of (4) 100−r=0.4,8,12,......,96,100 When 100−r is a multiple of 5 100−r=0.5.10,.......,100 ∴r=0,4,8,12,.......,96,100 and r=0.5,10.100 The rational number terms comes out to be. r=0.20.40,60,80,100 Thus, there are 6 rational terms.