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# Powers of complex numbers

## Video transcript

so I have the complex number cosine of 2 pi over 3 or 2/3 pi plus I sine of 2/3 PI and I'm going to raise that to the 20th power and what I want to do is first plot this number in blue on the complex plane and then figure out what it is raised to the 20th power and then try to plot that and I encourage you to pause this video and try this out on your own before I work through it so let's first flow focus on this blue complex number right over here it's clearly written in polar form and the angle is 2/3 PI 2/3 PI or 2 over or 2 PI over 3 radians and it's and it's and it's magnitude of this complex number is clearly 1 and to make it a little clearer you could write it in kind of the pure polar form where you have its magnitude out front so it's cosine of 2 over 3 pi plus I sine of 2 over 3 pi so you could write it just like that and when you look at that the angle is 2 over 3 pi and so that would get us let's see this is 0 this is PI we're going to go 2/3 of the way to PI each of these is 1 2 3 4 5 6 7 8 9 10 11 12 each of these is pot is is PI over 12 so we're going to go two-thirds of the way would be 8 PI's over 12 so 1 2 3 4 5 6 7 8 and the way I was able to reason through that is that 2/3 pi is the same thing as 8 PI over 12 each of these segments is PI over 12 so I just counted eight of them so that's that number but now let's try to raise it now let's try to raise it to the 20th power let me clear this and to do that we're going to use Euler's formula and Oilers formula you might remember tells us that e to the I theta is equal to cosine of theta plus I sine plus I sine of theta and so you see right over here this is already written in that form where theta is 2/3 pi so we can rewrite what we have in blue here as e let me write over here as e to the 2/3 PI I and then of course we're raising that to the 20th power we're raising that to the 20th power and this simplifies things dramatically because here if I try to multiply this thing time in factor 20 of these things and I multiply them together that would get really really really hairy really fast but here I can just use exponent properties this is going to be the same thing as e to the if ice raise something to exponent and then raise that to an exponent I can just take the product of the exponents so this is e to the 20 times 2 over 3 PI I which is equal to e to the 40 over 40 over 3 PI I now this is this number raised to the 20th power but this is a awfully large angle right over here I mean if we're thinking 40 over 3 PI let's just try to digest this 40 over 3 pi this is the same thing as this is the same thing let's see 40 divided by 3 is 13 and 1/3 this is the same thing as 13 and 1/3 times pi and we know that going to pi 2 pi radians gets you around the unit circle once so this is going over 6 times around the unit circle to get to or around I should say not the unit serve or going 6 times around the I guess going in circles in order to get to the point we want to so in order to simplify this a little bit let me subtract the largest multiple of 2 pi that I could figure that to kind of get this in a small of a form as possible because we know an angle if we have some angle it's equal to that angle plus some multiple of 2 pi where K is any integers okay could also be negative we could be subtracting a multiple of 2 pi so let me subtract let's see the largest multiple of 2 pi that I could subtract here is going to be 12 pi so let me subtract 12 from this so if I subtract 12 pi so I'll do it down here so 13 and 1/3 PI minus 12 pi remember I'm just trying to subtract the largest multiple of 2 pi that I can so minus 12 pi is going to move up so I rewrote PI minus 12 PI there 13 and 1/3 minus 12 is 1 and 1/3 that's going to be 1 and 1/3 pi or we could write it as 4/3 pi so this this complex number is going to be equivalent to e to the 4/3 pi I and this makes it much simpler and much easier for me to plot so 4/3 pi or the same thing as 1 and 1/3 pi so this would be PI and now we have to just go another 1/3 pi and each of these are 12 so if we go for 12 PI so each of these I sorry each of these are PI over 12 so go 4 PI over 12 so 1 2 3 4 gets us right over there and so this is so this number is the 20th power is this which is equivalent to this which we've plotted right over there now what if we wanted to take it to let's say the 21st power well then we would then we would increase the angle by another 2 PI over 3 or 8 PI over 12 so we would increase the angle by 1 2 3 4 5 6 7 8 and we would go right over there so how does this make conceptual sense well the number to the first power was right over here that was our original number in blue right over here if you raise it to the second power then you're increasing the angle by 2/3 pi you're increasing the angle to go there your e to the third power you increase the angle by 2/3 pi you go over there fourth power you get back here fifth sixth seventh eighth ninth tenth eleventh 12 13 14 15 16 17 18 19 20 a--the power gets us right over there