okay in this video I'm gonna show you guys how to find all the cube roots of the number
one and you might be wondering isn't the answer just equal to one yes if you just want to
talk about the real numbers but no if you are talking about the complex numbers as well
let me show you how and I will show you guys two ways to do this the first way is just
going to be algebra and in the second way is that we will have to use sine cosine and
also the polar form of a complex number to help us out okay so take a look and by the
way this is also called the the cube root of unity the unity is the number one okay
anyway here we go to find the cube root of a number is that I want to have a number X
so that when I raise that to the third power I 1 doubles the number I'm looking for namely
1 so we have to solve this equation and as I mentioned earlier one's definitely a solution
for this right because once plus their powers you could want and to find out the other students
this is what we can do let's minus 1 on both sides so I will have X to the third power
on the Left minus 1 but you see this is going to be a difference of a cube and I'm going
to put 1 as 1 to the 3rd power as well so I can look at this as a difference of two
cubes okay and then of course we have this equal to 0 now we can factor this out and
to factor this out the first factor is X minus 1 like this and then the second one is going
to be X and you square that so you have x squared and you add it with this times that
x times 1 just X and then you add in which 1 squared is still 1 so this is what we have
and now let's just do a little business put this down as equal to 0 so X minus 1 is equal
to 0 and then of course X is equal to 1 and to find out the complex answers here is the
deal I will have to make x squared plus X plus 1 equal to 0 and to solve this that's
just use the quadratic formula and notice that the aapc values are whole equal to 1
it's really cool huh anyway so I will just tell you that X its equal to negative B which
is 1 plus minus square root of B squared which is all right down P Square 1 square right
and the minus 4 times 8 which is 1 and C is also wise or four times one times one and
then all over two times one right so this is what we have and now of course we can simplify
this a little bit this is negative one plus minus square root this is one minus this is
4 1 minus 4 is negative 3 like this and then over 2 times 1 which is 2 okay so in the end
I will just tell you guys here are these three answers the first one is the real answer X
is equal to 1 and then I will write down the second answer for you guys this way I will
tell you the second is X equal to negative 1/2 let's put that down first because you
want to brag complex number in the a plus bi form right usually speaking anyway I'll
put down plus and you know you have a negative the square root so that's the I and then you
have the square root of 3 over 2 okay and then X equal to negative 1 over 2 and then
we have the minus version and then we'll have the I and then the square root of 3 over 2
like that and you can put I here or put the I at the end right here it's like up to you
seriously okay bye anyway here are the three answers for you guys now here is the more
interesting part and perhaps it's easier if you have seen the sine cosine and also the
polar form of a complex number but you know if you haven't seen this then know you can
just pick something up as well that's really cool so anyway here is the deal to find a
cube root of 1 we are going to look at I know we have some complex answers right so I will
use Omega you can use Z as well okay that's a really matter anyway the Omega to a third
power we want this to be equal to 1 in this case Omega it's a complex number and I will
write Omega as a plus B I okay and now the idea is that when we have a complex number
and once we raise that to a power it pretty much just rotates especially that talking
about a unit like this okay so now idea is that I will be looking at the complex plane
right okay so here is the real axis and this is the imaginary axis axis anyway Omega 2
star is equal to 1 that's a justice to look at a unit circle right a circle with radius
1 okay anyway I want to and now it's one the idea is that I want to have the first of limited
to down this is like 0 plus R 1 plus 0i so we're talking about this pond right here okay
so what this is 1 plus Zi Rui okay and then I want to go from here you know that's the
first one I want to go from here I want to rotate rotate rotate and a compact here okay
that's the deal well the idea is that a circle is drawn just 60 degrees I want to rotate
rotate rotate Minya do his three times 360 degrees divided by 3 is 120 each time right
and yes I'm just going to be using degrees okay anyway so if you go from here let me
just put this down let me just make this dark first this is starting and then rotate 120
degrees you got talking about somewhere here right so let me just draw this for you guys
120 degrees and guess what if you can find out the coordinate of this pond right here
you have the cube root of 1 okay you have a solution to this and then we can do the
same again right so if you start right here and then rotate 120 degrees like this in fact
you pretty much just do a reflection about the x-axis and let me just draw this focus
down like this so from here to here is another 120 degrees and this point it's also going
to be representing a cube root of 1 and our goal is to figure out the quotient of this
point and that point this points release is just 1 plus u 4i by the way to do so you see
that we have the ink already right so I'm going to look at this I will change that into
the pole of warm and to do that the x value is pretty much the x value right you just
associate that with it's the usual polar coordinate it's just that we will have our times cosine
or forever anger that you're talking about so put on later for that and then I will put
on the plus the B is just a white value so your sign and you still have the R and let
me put down the I first okay and then we have R sine theta like this so this is pretty much
it and in our case R is equal to what okay r is equal to 1 and you can imagine if you
have Omega to the third power is equal to 8 then in this case the R will be what the
R will be two okay yeah you can try it yeah because you want to look here R as the real
value to a third power is equal to 8 in that case so r is equal to 2 pi anyway in our case
this is nice r is equal to 1 okay and now you have the answers pretty much the first
answer i'll Jojo Omega is equal to 1 and then the second answer is going to be Omega equal
to because ours one already so let me just put on the cosine and also the sine part and
I of course first one is cosine of 120 degrees and yes I am still using degrees but hopefully
that doesn't bother you I and ours one so we have I times sine and then theta is once
again 120 degrees and then the last one is going to be Omega equals two cosine of 240
degrees right because you have to go from here to here okay this is a total of 240 degrees
like that so 240 degrees right here plus I sine of 240° and if you want to know more
about this you know I have some pages where it gets to be over you can also talk about
that the mauryas theorem for this kind of things and prenuptial rotation but anyway
I just want to get this done for you guys first answer Omega equals to 1 now cosell
120 if you want to do some usual trigonometry things you can look at a reference triangle
right here with this n gulping 60 degrees right and anyway Cosell 120 degrees is going
to be negative 1/2 so the second is going to be right here we have negative 1/2 plus
signing off 120 degrees it's going to be square root of 3 over 2 so I will put down square
root 3 over 2 with the eye in front just like that and now for this one Omega will be cosine
of 240 they have the same X values so we still have negative 1/2 but this time sine of 240
degrees will have the negative value of this white value so it's become minus minus I square
root of 3 over 2 like this and with this we are done okay in fact I'm gonna use this in
one of my videos later on so that's what I'm doing this but hopefully you guys also fine
now this is pretty cool and leave a comment down below and let me know which method do