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