Question

Is (1+i14+i18+i22) a real number justify your answer?

Answer 1

Answer:

(1 +  i¹⁴ + i¹⁸ + i²²) is real

Step-by-step explanation:

We are given that

(1 +  i¹⁴ + i¹⁸ + i²²)

We know that

Now

i¹⁴ = i¹²⁺² = i¹² × i²              ∵ i¹²⁺² = i¹² × i²

   = (i⁴)³ × i²

   = (1)³ × i² = - 1                ∵ i² = -1 ⇒ i⁴ = 1

Similarly

i¹⁸ = i¹⁶⁺²

   = i¹⁶ × i²

   = (i⁴)⁴ × i²

   = (1)⁴ × i² = - 1                ∵ i² = -1 ⇒ i⁴ = 1

i²² = i²⁰⁺²

    = i²⁰ × i²

    = (i⁴)⁵ × i²

    = (1)⁵ × i² = -1                  ∵ i² = -1 ⇒ i⁴ = 1

By putting values in our given expression we get

(1 +  i¹⁴ + i¹⁸ + i²²) = ( 1 - 1 - 1 - 1)

                           = -2

And we know that -2 is a real number

So

(1 +  i¹⁴ + i¹⁸ + i²²) is real