Question -
Answer -
(i)┬аi┬а457
Let us simplify weget,
i457┬а=i┬а(456 + 1)
= i┬а4(114)┬а├Чi
= (1)114┬а├Чi
= i [since i4┬а=1]
(ii)┬аi┬а528
Let us simplify weget,
i┬а528┬а=i┬а4(132)
= (1)132
= 1 [since i4┬а=1]
(iii)┬а1/ i58
Let us simplify weget,
1/ i58┬а=1/ i┬а56+2
= 1/ i┬а56┬а├Чi2
= 1/ (i4)14┬а├Чi2
= 1/ i2┬а[since,i4┬а= 1]
= 1/-1 [since, i2┬а=-1]
= -1
(iv)┬аi┬а37┬а+1/i┬а67
Let us simplify weget,
i┬а37┬а+1/i┬а67┬а= i36+1┬а+ 1/ i64+3
= i + 1/i3┬а[since,i4┬а= 1]
= i + i/i4
= i + i
= 2i
(v)┬а[i┬а41┬а+1/i┬а257]
Let us simplify weget,
[i┬а41┬а+1/i┬а257] = [i40+1┬а+ 1/ i256+1]
= [i + 1/i]9┬а[since,1/i = -1]
= [i тАУ i]
= 0
(vi)┬а(i┬а77┬а+i┬а70┬а+ i┬а87┬а+ i┬а414)3
Let us simplify weget,
(i┬а77┬а+i┬а70┬а+ i┬а87┬а+ i┬а414)3┬а=(i(76 + 1)┬а+ i(68 + 2)┬а+ i(84 + 3)┬а+i(412 + 2)┬а)┬а3
= (i + i2┬а+i3┬а+ i2)3┬а[since i3┬а=тАУ i, i2┬а= тАУ 1]
= (i + (тАУ 1) + (тАУ i) +(тАУ 1))┬а3
= (тАУ 2)3
= тАУ 8
(vii)┬аi┬а30┬а+i┬а40┬а+ i┬а60
Let us simplify weget,
i┬а30┬а+i┬а40┬а+ i┬а60┬а= i(28 + 2) +┬аi40┬а+i60
= (i4)7┬аi2┬а+(i4)10┬а+ (i4)15
= i2┬а+110┬а+ 115
= тАУ 1 + 1 + 1
= 1
(viii)┬аi┬а49┬а+i┬а68┬а+ i┬а89┬а+ i┬а110
Let us simplify weget,
i┬а49┬а+i┬а68┬а+ i┬а89┬а+ i┬а110┬а=i(48 + 1)┬а+ i68┬а+ i(88 + 1)┬а+ i(116+ 2)
= (i4)12├Чi+ (i4)17┬а+ (i4)22├Чi + (i4)29├Чi2
= i + 1 + i тАУ 1
= 2i