Question 79349
Here we are given the values of i till {{{i^8}}} = 1 


Now to find {{{i^65}}} this can be written as: 


This can be written as: 


{{{(i^13)^5}}} 



Lets now find the value of {{{i^13}}} from the given data: 


{{{i^8}}} = 1 


{{{i^9}}} =  {{{i^8 * i}}} 


          = {{{1 * i}}} 


         = i 


{{{i^10}}} = {{{i^9 *i}}}  


           =  i * i 

           
           = {{{i^2}}}


           = -1  


Similarly the rest are also found out. 


{{{i^11}}} = -i 



{{{i^12}}} = 1 


{{{i^13}}} = i 


Thus the value of {{{i^13}}} = i


And we know that {{{i^5}}}  = i 



Thus, {{{i^65}}} = {{{(i^13)^5}}} 


                 = {{{(i)^5}}} 


                 = i 


Hence, the solution. 



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 2nd method



Or we can also find the value of {{{i^13}}} = {{{((i^6)^2) *i}}} 


                                            = {{{((-1)^2)*i}}} 


                                            = {{{1 * i}}} 


                                            = i 


Now 65  = 13 * 5 


{{{i^65}}} = {{{(i^13)^5}}} 


          = {{{(i)^5}}} 


          = i 





Hence, the solution.


Regards

Chitra
Online MAth tutor
www.knowledgeonlineservices.com