Answer
a) i¹¹³ = i
b) i⁸² = -1
c) i⁴⁸ = 1
d) i²²³ = -1
Explanation
The complex number i, is expressed mathematically as
i = √(-1)
So,
i² = i × i = √(-1) × √(-1) = -1
i³ = i²× i = -1 × i = -i
i⁴ = i² × i² = -1 × -1 = 1
Hence, we can see that when the power on the complex number (i) is a multiple of 4, the answer will always be equal to 1.
So, this is the basis to solving all the expressions given.
a) i¹¹³
For 113, the closest multiple of 4 to this number is 112.
i¹¹³ = i¹¹² × i
i¹¹² = 1
i¹¹³ = i¹¹² × i = 1 × i = i
b) i⁸²
The closest multiple of 4 to the number 82, is 80.
i⁸² = i⁸⁰ × i²
i⁸⁰ = 1
i² = -1
i⁸² = i⁸⁰ × i² = 1 × -1 = -1
c) i⁴⁸
The number 48 is a multiple of 4, so,
i⁴⁸ = 1
d) i²²³
For the number 223, the closest multiple of 4 to this number is 220.
i²²³ = i²²⁰ × i³
i²²⁰ = 1
i³ = -i
i²²³ = i²²⁰ × i³ = 1 × -i = -i
Hope this Helps!!!
