How do you simplify `i^13` ?

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lemjay | High School Teacher | (Level 3) Senior Educator

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To simplify `i^n` , divide the exponent by 4. Then, consider only the remainder of the quotient.

  >> If remainder is 0 ,then `i^n=1` . 

  >> If reminder is 1, then `i^n=i` .

 >> If remainder is 2, then `i^n=-1` .

 >> If remainder is 3, then `i^n=-i` .

Applying this, the remainder of `i^13` is:

13 `-:`  4 = 3 Remainder 1

Hence, `i^13=i` .

Another way is to breakdown `i^13` to factors of `i^4` .

`i^13=i^4*i^4*i^4*i=(i^4)^3*i`

Since `i^4=1` , then

`=(1)^3*i=1*i=i`

Thus, either method results to `i^13=i` .

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