Question

imaginary number pattern equation?
How can I simplify this without expanding it the long way...?
(i^68)

Answer 1

what is i^2?

Answer 2

-1

Answer 3

use rules of exponents to change i^68 to (i^2)^something

Answer 4

but how would that work

Answer 5

like I know youcan do (i^17)^4 but how do you know the answer is 1

Answer 6

One shortcut is to divide 68 by 4 to get a quotient and a remainder

if the remainder is 0, then i^68 = 1
if the remainder is 1, then i^68 = i
if the remainder is 2, then i^68 = -1
if the remainder is 3, then i^68 = -i

Answer 7

ah. change it around to (i^4)^17
what is i^4? what is (-1)^2?

Answer 8

OH THAT IS SO COOL LOL @jim_thompson5910 THNX

Answer 9

yw

Answer 10

@jim_thompson5910 wait but what if its not divisible by 4

Answer 11

that's why we only care about the remainder

Answer 12

OH ahha.. I see thanks

Answer 13

for example,

i^9 = i because 9/4 = 2 remainder 1

i^15 = -i since 15/4 = 3 remainder 3

i^60 = 1 since 60/4 = 15 remainder 0

we're only looking at the remainders when we're using that rule

Answer 14

alright I understand it thanks!

Answer 15

np