Question -
Answer -
(i) i 457
Let us simplify weget,
i457 =i (456 + 1)
= i 4(114) ×i
= (1)114 ×i
= i [since i4 =1]
(ii) i 528
Let us simplify weget,
i 528 =i 4(132)
= (1)132
= 1 [since i4 =1]
(iii) 1/ i58
Let us simplify weget,
1/ i58 =1/ i 56+2
= 1/ i 56 ×i2
= 1/ (i4)14 ×i2
= 1/ i2 [since,i4 = 1]
= 1/-1 [since, i2 =-1]
= -1
(iv) i 37 +1/i 67
Let us simplify weget,
i 37 +1/i 67 = i36+1 + 1/ i64+3
= i + 1/i3 [since,i4 = 1]
= i + i/i4
= i + i
= 2i
(v) [i 41 +1/i 257]
Let us simplify weget,
[i 41 +1/i 257] = [i40+1 + 1/ i256+1]
= [i + 1/i]9 [since,1/i = -1]
= [i – i]
= 0
(vi) (i 77 +i 70 + i 87 + i 414)3
Let us simplify weget,
(i 77 +i 70 + i 87 + i 414)3 =(i(76 + 1) + i(68 + 2) + i(84 + 3) +i(412 + 2) ) 3
= (i + i2 +i3 + i2)3 [since i3 =– i, i2 = – 1]
= (i + (– 1) + (– i) +(– 1)) 3
= (– 2)3
= – 8
(vii) i 30 +i 40 + i 60
Let us simplify weget,
i 30 +i 40 + i 60 = i(28 + 2) + i40 +i60
= (i4)7 i2 +(i4)10 + (i4)15
= i2 +110 + 115
= – 1 + 1 + 1
= 1
(viii) i 49 +i 68 + i 89 + i 110
Let us simplify weget,
i 49 +i 68 + i 89 + i 110 =i(48 + 1) + i68 + i(88 + 1) + i(116+ 2)
= (i4)12×i+ (i4)17 + (i4)22×i + (i4)29×i2
= i + 1 + i – 1
= 2i