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Note the pattern

i^1  = i

i^2  = - 1

i^3  = -i

i^4  = 1

So....all  we need to do is to divide  each exponent by 4, thusly

22/4  =  5 +  2/4      the numerator of the fraction = 2..... tells us where  i^22 falls in the pattern  =  -1

And....likewise

222/4 = 55 + 2/4     =   -1

So

i^22 + i^222   =

-1 + -1  =

-2