Roots of Unity? (1 Viewer)

[ashokkumar]

Can someone explain how to do this question?


1, w and w^2 are the three cube roots of unity. State the values of w^3 and 1 + w + w^2. Hence simplify each of the expressions (1 + 3w + w^2)^2 and (1 + w + 3w^2)^2 and show that their sum is -4 and their product is 16.

(also, why do we call them roots of unity?)

 

[ninetypercent]

unity meaning 1. roots of unity = roots of one

if w is a cube root of unity, then
w^3 = 1

w^3 -1 = 0
(w-1)(w^2 + w +1) = 0
(w^2 + w +1) = 0 (if w is not equal to 1)

(1 + 3w + w^2)^2 = (-w + 3w)^2 = (2w)^2 = 4w^2

(1 + w + 3w^2)^2 = (-w^2 + 3w^2)^2 = (2w^2)^2 = 4w^4 = 4w

sum of them = 4w + 4w^2 = 4(w+w^2) = 4(-1) = -4

product of them = 4w^2(4w) = 16w^3 = 16

 

[fullonoob]

(1 + 3w + w^2)^2 = (-w + 3w)^2 = (2w)^2 = 4w^2

(1 + w + 3w^2)^2 = (-w^2 + 3w^2)^2 = (2w^2)^2 = 4w^4 = 4w

can please explain this step, i don't see it

 

[fullonoob]

rofl dw got it now LOL
but dont get where you obtain (-w + 3w)^2 from
i just did since (w^2 + w +1) = 0
(0+2w)^2 = 4w^2
same for the other one

 

[bachviete]

(1 + 3w + w^2)^2 = (-w + 3w)^2 = (2w)^2 = 4w^2

(1 + w + 3w^2)^2 = (-w^2 + 3w^2)^2 = (2w^2)^2 = 4w^4 = 4w

can please explain this step, i don't see it

You have to know these two rules





ie


These are explained in both Excel MX2 and Cambridge (i think)

So



and




EDIT: See you got it =)

 

[fullonoob]

You have to know these two rules





ie


These are explained in both Excel MX2 and Cambridge (i think)

So



and




EDIT: See you got it =)

umm thats not really my question
(1 + 3w + w^2)^2 = >>>>>(-w + 3w)^2<<<< = (2w)^2 = 4w^2

(1 + w + 3w^2)^2 = >>>>(-w^2 + 3w^2)^2<<<< = (2w^2)^2 = 4w^4 = 4w
where does he get this. Probs another method but dno what it is

 

[bachviete]

umm thats not really my question
(1 + 3w + w^2)^2 = >>>>>(-w + 3w)^2<<<< = (2w)^2 = 4w^2

(1 + w + 3w^2)^2 = >>>>(-w^2 + 3w^2)^2<<<< = (2w^2)^2 = 4w^4 = 4w
where does he get this. Probs another method but dno what it is

Allow me to explain ;P lol

 

[fullonoob]

Allow me to explain ;P lol

lol you know i actually saw this but couldn't be bothered subbing it back in to check xD
thanks anyways