Let w be a solution of x²+x+1 = 0
What is the value of w^10 + w^5 + 3?

Question

Let w be a solution of x²+x+1 = 0
What is the value of w^10 + w^5 + 3?

chillout · Answer

w is$$\frac{i\sqrt{3}}{2}-1/2.$$ Is there a simple way to calculate w^10 and w5 other than doing lots of multiplications?

chillout · Answer

Oh, and it can be its conjugate too.

fifciol · Answer

substitute p= w^5

chillout · Answer

All right... Rewriting it, p² + p + 3. But what I asked was how can I make the multiplication not too tiring?

fifciol · Answer

solve for p and you will have two solutions equal to w^2, square them and substitute in equation of w

anonymous · Answer

Well the cube root of 1 is w, and there is a equation that says 1+w+w^2=0 So, from here x= w Now W^10=(w^2)^5= (-(1+w))^5 From here you might be able to calculate.

fifciol · Answer

w^5 *

chillout · Answer

@kutabs , wouldn't it be that w³ is 1?

chillout · Answer

Oh right, I got it. The answer is 2, isn't it? THanks for the help.

anonymous · Answer

@Chillout :Of course.
w^3=1
Or, w^3-1=0
Or, (w-1)(w^2+w+1)=0         (From a^3-b^3)
Either, the roots are  w=1 or 1+w+w^2=0

anonymous · Answer

Yeah, 2. :)