Evaluate the following:
(i) i457
(ii) i528
(iii)
(iv)
(v)
(vi) (i77 + i70 + i87 + i414 )3
(vii) (vii) i30 + i40 + i60
(viii) i49 + i68 + i89 + i118
i. i457 = i (456 + 1)
= i4(114) × i
= (1)114 × i = i since i4 = 1
ii. i528 = i4(132)
= (1)132 =1 since i4 = 1
iii.
since i4 = 1
= – 1 since i2 = – 1
iv.
[since i4 = 1]
v.
= (i – i) = 0
[since ]
vi. (i77 + i70 + i87 + i414 )3 = (i(76 + 1) + i(68 + 2) + i(84 + 3) + i(412 + 2) ) 3
(i77 + i70 + i87 + i414 )3 = (i + i2 + i3 + i2 )3
[since i3 = – i, i2 = – 1]
= (i + (– 1) + (– i) + (– 1))3 = (– 2)3
(i77 + i70 + i87 + i414 )3 = –8
vii. i30 + i40 + i60 = i(28 + 2) + i40 + i60
= (i4)7 i2 + (i4)10 + (i4)15
= i2 + 110 + 115 = – 1 + 1 + 1 = 1
viii. i49 + i68 + i89 + i118 = i(48 + 1) + i68 + i(88 + 1) + i(116 + 2)
= (i4)12×i + (i4)17 + (i4)11×i + (i4)29×i2
= i + 1 + i – 1 = 2i