Simplify the power of i i i . i 1003 i^{1003} i 1003

Hey, y'all. Let's take a look at this practice problem. So here I have I to the power of 1,003, and I wanna simplify this power of I. So I'm gonna ask myself, is this 1,003 evenly divisible by 4? And my answer is no, so that means I need to find my remainder of dividing by 4. So let's go ahead and take 1,003 and divide it by 4 using long division. So 4 doesn't go into 1 at all, but it does go into 10 two times. So I have that 8. 10 minuteus 8 will give me 2, and then I have 20. 4 goes into 20 exactly 5 times. So if I take 20 20, don't have anything there. I need to pull my 3 down. And 4 cannot go into 3 at all, so it goes in 0 times. And I am simply left with 3 as my remainder. So that means that I to the power of 1,003 is really just I cubed. So our remainder here is 3 I cubed, which just gives me negative I. So I to the 1,003rd power is just negative I. Thanks for watching, and I'll see you in the next one.