Question

Simplify: i^41

1) i
2) -1
3) -i
4) 1

Answer 1

1) i

Answer 2

Simplify: i^72

1) i
2) -1
3) -i
4) 1

Answer 3

4) 1

Answer 4

i
i^2=-1
i^3=i*i^2=i*(-1)=-i
i^4=i^2*i^2=1
i^5=i^4*i=1*i=i
i^6=i^5*i=i*i=i^2=-1
this is gonna keep repeating it self
i,-1,-i,1,i,-1,-i,1, and so on...
the trick is to take the 72 and figure out the remainder when 72 is divided by 4

Answer 5

So would this answer i or 1?

Answer 6

18
______
4| 72
-4
----
32
-32
----
0

the remainder is 0

so you have
\[i^{72}=i^{4(18)+0}=(i^4)^{18}=(1)^{18}=1\]
----------------------
if you have \[i^a\]
\[i^a=i^{4(k)+j}\]
if j=0, then answer is 1
if j=1, then answer is i
if j=2, then answer is -1
if j=3, then answer is -i

Answer 7

assuming a=4k+j of course

Answer 8

So the answer is #1?

Answer 9

if you read what i have then you will find the answer
i did it for you and so did euler
i was just showing work