Answer:
[tex]i^{15} = -i[/tex]
Step-by-step explanation:
The basic relation of the complex numbers is:
[tex]i^{2} = -1[/tex]
So, we decompose [tex]i^{15}[/tex] in factors of [tex]i^{2}[/tex].
So:
[tex]i^{15} = i^{2}*i^{2}*i^{2}*i^{2}*i^{2}*i^{2}*i^{2}*i[/tex]
Each [tex]i^{2}[/tex] is replaced by -1.
So:
[tex]i^{15} = (-1)*(-1)*(-1)*(-1)*(-1)*(-1)*(-1)*i[/tex]
[tex]i^{15} = (-1)^{7}*i[/tex]
Any negative value powered to an odd value will be negative. So:
[tex](-1)^{7} = -1[/tex]
[tex]i^{15} = -i[/tex]
