Question

Evaluate the following:

(i) i⁴⁵⁷

(ii) i⁵²⁸

(iii) 1/i⁵⁸

(iv) i³⁷ + 1/i⁶⁷

(v) [i⁴¹ + 1/i²⁵⁷]

(vi) (i⁷⁷ + i⁷⁰ + i⁸⁷ + i⁴¹⁴)³

(vii) i³⁰ + i⁴⁰ + i⁶⁰

(viii) i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰

Answer 1

(i) i⁴⁵⁷

Now let us simplify we get,

i⁴⁵⁷ = i^{(456 + 1)}

= i⁴⁽¹¹⁴⁾ × i

= (1)¹¹⁴ × i

= i[since i⁴ = 1]

(ii) i⁵²⁸ 

Now let us simplify we get,

i⁵²⁸ = i⁴⁽¹³²⁾

= (1)¹³² 

= 1[since i⁴ = 1]

(iii) 1/i⁵⁸

Now let us simplify we get,

1/i⁵⁸ = 1/i^{56 + 2}

= 1/i⁵⁶ × i²

= 1/(i⁴)¹⁴ × i²

= 1/i² [since, i⁴ = 1]

= 1/-1 [since, i² = -1]

= -1

(iv) i³⁷ + 1/i⁶⁷

Now let us simplify we get,

i³⁷ + 1/i⁶⁷ = i^{36 + 1} + 1/i^{64 + 3}

= i + 1/i³ [since, i⁴ = 1]

= i + i/i⁴

= i + i

= 2i

(v) [i⁴¹ + 1/i²⁵⁷]

Now let us simplify we get,

[i⁴¹ + 1/i²⁵⁷] = [i^{40 + 1} + 1/i^{256 + 1}]

= [i + 1/i]⁹ [since, 1/i = -1]

= [i – i]

= 0

(vi) (i⁷⁷ + i⁷⁰ + i⁸⁷ + i⁴¹⁴)³

Now let us simplify we get,

(i⁷⁷ + i⁷⁰ + i⁸⁷ + i⁴¹⁴)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) i³⁰ + i⁴⁰ + i⁶⁰

Now let us simplify we get,

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 + i¹¹⁰

Now let us simplify we get,

i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰ = 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