Answer:
-1
Step-by-step explanation:
I simplified the expression using the definition of an imaginary number i=√1
First I applied the exponent rule: [tex]a^{bc}=\left(a^b\right)^c[/tex]
[tex]i^{82} = \left(i^2\right)^{41}[/tex]
[tex]=\left(i^2\right)^{41}[/tex]
Then, I applied the maginary number rule: [tex]\:i^2=-1[/tex]
[tex]=\left(-1\right)^{41}[/tex]
Then I used the exponent rule again: [tex]\left(-a\right)^n=-a^n,\:\mathrm{if\:}n\mathrm{\:is\:odd}[/tex]
[tex]\left(-1\right)^{41}=-1^{41}[/tex]
[tex]=-1^{41}[/tex]
Sooo, [tex]1^{41}=1,[/tex] which equals =-1.
