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Question -

Evaluate the following:

(i) i┬а457

(ii) i┬а528

(iii) 1/ i58

(iv) i┬а37┬а+ 1/i┬а67

(v) [i┬а41┬а+ 1/i┬а257]

(vi) (i┬а77┬а+ i┬а70┬а+ i┬а87┬а+i┬а414)3

(vii) i┬а30┬а+ i┬а40┬а+ i┬а60

(viii) i┬а49┬а+ i┬а68┬а+ i┬а89┬а+i┬а110



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

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