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Question

Evaluate the following:
(i) i457
(ii) i528
(iii) 1i58
(iv) i37+1i67
(v) i41+1i2579
(vi) (i77+i70+i87+i414)3
(vii) i30+i40+i60
(viii) i49+i68+i89+i110

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Solution


i i457 =i4×114+1 =i4114×i =i i4=1ii i528 =i4×132 =i4132 =1 i4=1iii 1i58=1i4×14+2 =1i414×i2 =1i2 i4=1 =-1 i2=-1iv i37+1i67=i4×9+1+1i4×16+3 =i49×i+1i416×i3 =i-1i i3=-i =i-1i×ii =i-ii2 =i--i i2=-1 =2i v i41+1i2579 =i4×10+1+1i4×64+19 =i410×i+1i464×i9 =i+1i9 i4=1

=i+ii29 =i-i9 i2=-1=0

vi i77+i70+i87+i4143=i4×19+1+i4×17+2+i4×21+3+i4×103+23 =i419×i+i417×i2+i421×i3+i4103×i2 =i-1-i-13 i4=1,i3=-i and i2=-1 =-23 =-8vii i30+i40+i60=i4×7+2+i4×10+i4×15 =i47×i2+i410+i415 =-1+1+1 i4=1,i2=-1 =1 viii i49+i68+i89+i110=i4×12+1+i4×17+i4×22+1+i4×27+2 =i412×i+i417+i422×i+i427×i2 =i+1+i-1 i4=1,i2=-1 =2i

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