Evaluate : `(i) i^(23)" "(ii) i^(998)" "(iii)i^(-998)" "(iv) i^(-71)` `(v) (sqrt(-1))^(91)" "(vi) (i^(37)xx i^(-61))" "(vii) i^(-1)`

Evaluate : `(i) i^(23)" "(ii) i^(998)" "(iii)i^(-998)" "(iv) i^(-71)`

1.	Evaluate : `(i) i^(23)" "(ii) i^(998)" "(iii)i^(-998)" "(iv) i^(-71)` `(v) (sqrt(-1))^(91)" "(vi) (i^(37)xx i^(-61))" "(vii) i^(-1)`
Answer» We have `(i)" "i^(23) = i^((4xx5)+3)=(i^(4))^(5)xx i^(3) = i^(3) = -i." "[because i^(4)=1]` `(ii)" "i^(998) = i^(4xx249+2)=(i^(4))^(249)xx i^(2) = (1 xx i^(2)) = i^(2) = -1." "[because i^(4)=1]` `(iii)" "i^(-998) = (1)/(i^(998))xx(i^(2))/(i^(2))=(i^(2))/(i^(1000))=(-1)/(1)= -1." "[because i^(1000)=(i^(4))^(250)=1]` `(iv)" "i^(-71) = (1)/(i^(71))xx(i)/(i)=(i)/(i^(72))=(i)/((i^(4))^(18))=(i)/(1)= i." "[because i^(4)=1]` `(v)" "(sqrt(-1))^(91)= i^(91) = i^(4 xx 22 + 3) = (i^(4))^(22)xx i^(3) = 1 xx(-i) = -i." "[because i^(3)=i]` `(vi)" "i^(37) = i^(4 xx 9 + 1) = (i^(4))^(9) xx i = 1 xx i = i`. `" "i^(-61)=(1)/(i^(61))xx(i^(3))/(i^(3))=(i^(3))/(i^(64))=(-i)/((i^(4))^(16))=(-i)/(1)=-i`. `" "therefore" "(i^(37)+i^(-61))=i+(-i)=0`. `(vii)" "i^(-1)=(1)/(i)=(1)/(i)xx(i^(3))/(i^(3))=(-i)/(i^(4))=(-i)/(1) = -i`.