OpenStudy (anonymous):

simplify \[i ^{67}\]

OpenStudy (anonymous):

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

OpenStudy (amistre64):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

i got 16.74

OpenStudy (anonymous):

i meen i got 16.75

OpenStudy (anonymous):

hold the phone

OpenStudy (anonymous):

=i^3=-i

OpenStudy (amistre64):

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

OpenStudy (anonymous):

i mean divide, get whole number and take the remainder

OpenStudy (anonymous):

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

OpenStudy (anonymous):

i did not mean divide and get a decimal

OpenStudy (amistre64):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

so the answere is i or 1?

OpenStudy (anonymous):

its got to be 1 then

OpenStudy (anonymous):

slow

OpenStudy (amistre64):

67/4 = 16 R3. i^3 = i^0 = 1 i^1 = i i^2 = -1 i^4 = -i

OpenStudy (anonymous):

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

OpenStudy (amistre64):

.... i have cursed fingers lol

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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 \[i^0=1\]

OpenStudy (anonymous):

ok got it!

OpenStudy (anonymous):

good. clear what remainder means yes?

OpenStudy (zarkon):

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