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# Powers of the imaginary unit

CCSS.Math:

## Video transcript

now that we've seen that that as we take I to higher and higher powers it cycles between one I negative one negative I then back to one i negative one and negative I I want to see if we can tackle some I guess you could call them trickier problems and you might see these surface and they're also kind of fun to do to realize that you can use this the fact that I the powers of I cycle through these values you can use this to really on a back of an envelope take arbitrarily high powers of I so let's try just for fun let's see what I to the 100th power is and the realization here is that 100 is a multiple of four so you could say that this is the same thing as I to the I to the four times twenty fifth power and this is the same thing just from our exponent property this is I to the fourth power raised to the 25th power right if you have something raised to an exponent and then that is raised to an exponent that's the same thing as multiplying the two exponents and we know that I to the fourth that's pretty straightforward I to the fourth is just one I to the fourth is 1 so this is 1 so this is equal to one to the 25th power which is just equal to just equal to one so once again we use this kind of cycling ability of I when you take its powers to figure out a very high exponent of I now let's say we try something a little bit stranger let's try I to thee let's try it I to the 500 and first power now in this situation 501 it's not a multiple of four so you can't just do that that simply but what you could do is you could write this as a product of two numbers one that is a multiple one that is I to eight multiple of fourth power and that one it isn't and so you could rewrite this 500 is is a multiple of four so you could write this as I to the 500th power I to the 500th power times I to the first power right you have the same base when you multiply you can add exponents so this would be I to the five hundred and first power and we know that this is the same thing as I to the 500th power is the same thing as I to the fourth power four times what four times 100 95 is 500 so that's this part right over your eye of the 500th is the same thing as I to the fourth to the 125th power and then that times I to the first power times I to the first well I to the fourth is 1 1 to the 125th power is just going to be 1 this whole thing is 1 and so we are just left with we are just left with I to the 1st so this is going to be equal to I so it seems like a really daunting problem something that you would have to sit and do all day but you can use this cycling to realize look I to the 500th is just going to be 1 and so I to the 500th one is just going to be I times that so I to any multiple of four let me write this generally so if you have I to any multiple of four so this right over here is well we'll just restrict K to be non-negative right now K is greater than or equal to zero so if we have I to any multiple of four right over here we are going to get we are going to get one because this is the same thing as I to the fourth power to the K power and that is the same thing as 1 to the K power which is clearly equal to 1 and if we have anything else if we have I to the 4 K plus 1 power I've the 4 K plus 2 power we can then just do this technique right over here so let's try that with a few more problems just to make it clear that you can do really really arbitrarily crazy things so let's take I to the 7320 first power now we just have to figure out this is going to be some multiple of 4 top plus something else so to do that well you could just look at it by sight that 7320 is divisible by 4 you can verify that by hand and then you have that one left over and so this is going to be I to the 7320 times I to the first power this is a multiple of 4 this right here is a multiple 4 and I know that because any hundred is a multiple any thousand is a multiple of 4 and needs 100 is a multiple 4 and then 20 is a multiple of 4 and so this right over here will simplify to 1 sorry that's not I to the eighth power this is either the first power 7321 is 7320 plus one and so this part right over here is going to simplify to one and we're just going to be left with I to the first power or just I let's do another one I to the I to the 90 I to the ninety ninety let me try something interesting I to the 99th I to the 99th power so once again what's the highest multiple of four that is less than 99 it is 96 it is 96 so this is the same thing as I to the 96th power times I to the third power right if you multiply these same base add the exponent you would get I to the 99th power I to the 96th power since this is a multiple of four this is I to the fourth and then that to the 16th power so that's just 1 in the 16th so this is just 1 and then you're just left with I to the 3rd power and you could either remember that I to the third power is equal to you could just remember this equal to negative I or if you forget that you could just say look this is the same thing as I squared times I this is equal to I squared times I I squared by definition is equal to negative 1 so you have negative 1 times I is equal to is equal to negative I let me do one more just just for the fun of it let's take I to the 38th power well once again this is equal to I to the 36th times I squared I'm doing I to the 36 power since that's the largest multiple for that goes into 38 what's left over is this to this simplifies to one and I'm just left with I squared which is equal to negative one