Question

Question 1 Evaluate: [ i^18 + (1/i)^25 ] ^ 3

Class X1 - Maths -Complex Numbers and Quadratic Equations Page 112

Answer 1

[i^18 + (1/i)^25 ]³
we know,
i² = -1 and i² = -1
so, i^18 = i^(4×4+2)
= i^4×4.i²
= (i⁴)⁴.i²
=(1)⁴.i²
= 1.(-1)
= -1

again, (1/i) = i/i.i = i/-1 = -i
so, (1/i) ^25
= (-i)^25
= (-1)^25.(i) ^25
=(-1).i^(4×6+1)
= (-1). (i⁴)^6.i
= (-1)(1)^6.i
= -i

now,
[(i)^18 + (1/i) ^25 ]³ = [-1 -i]³
= (-1)³ + (-i)³ +3(-1)².(-i) +3.(-i)(-1)²
= -1 -i³ -3i +3
= 2 -2i
Hence, (2 -2i ) is the answer