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


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