Question

simplify \[i ^{67}\]

Answer 1

ok just like before. divide 67 by 4 and take the remainder

Answer 2

4|67 R... youre getting quicker ;)

Answer 3

lets call the remainder r
then
\[i^{67}=i^r\]

Answer 4

i got 16.74

Answer 5

i meen i got 16.75

Answer 6

hold the phone

Answer 7

=i^3=-i

Answer 8

its not 16.75 .. its either 1, -1, i or -i

Answer 9

i mean divide, get whole number and take the remainder

Answer 10

like if i divide 72 by 5 i get a remainder of 2

Answer 11

i did not mean divide and get a decimal

Answer 12

the reason we divide by 4 is that there is only 4 answers it can be; they just circle around on each other like the hands of a clock

Answer 13

if i divide 111 by 10 i get a remainder of 1
and if you divide 67 by 3 the remainder is...

Answer 14

so the answere is i or 1?

Answer 15

its got to be 1 then

Answer 16

slow

Answer 17

67/4 = 16 R3.

i^3 =

i^0 = 1
i^1 = i
i^2 = -1
i^4 = -i

Answer 18

\[i^1=i\]
\[i^2=-1\]
\[i^3=-i\]
\[i^4=1\]

Answer 19

.... i have cursed fingers lol

Answer 20

it is from this set
\[\{1, i, -1, -i\}\] depending on the remainder when you divide your exponent by 4

Answer 21

if the remainder is 1, you get
\[i^1=i\] if it is 2 you get
\[i^2=-1\] if 3 you get
\[i^3=-i\] and if 4 divides your exponent evenly you get
\[i^0=1\]

Answer 22

ok got it!

Answer 23

good. clear what remainder means yes?

Answer 24

\[i^{67}=i^{4\cdot16+3}=i^{4\cdot16}i^3=(i^4)^{16}i^3=1^{16}i^3=i^3\]