Given:
The complex number is [tex]i^{34}[/tex].
To find:
The value of given complex number.
Solution:
We have,
[tex]i^{34}[/tex]
It can be written as
[tex]i^{34}=i^{32+2}[/tex]
Using properties of exponents, we get
[tex]i^{34}=i^{32}i^{2}[/tex] [tex][\because a^{m+n}=a^ma^n][/tex]
[tex]i^{34}=i^{4\times 8}i^{2}[/tex]
[tex]i^{34}=(i^4)^8i^{2}[/tex] [tex][\because (a^{m})^n=a^{mn}][/tex]
We know that, [tex]i^4=1,i^2=-1[/tex].
[tex]i^{34}=(1)^8(-1)[/tex]
[tex]i^{34}=(1)(-1)[/tex]
[tex]i^{34}=-1[/tex]
The value of [tex]i^{34}[/tex] is -1. Therefore, the correct option is C.
