Given w^5 = 1 but w is not equal to 1. what is the value for x ? w^4 + w^3 + w^2 + w^1 + 1 = x?

Question

w^4 + w^3 + w^2 + w^1 + 1 = x
what is the value of x

Answer 1

most of persons have given correct ans but I would give it based on polynomials.

w^5 = 1
so w^5-1 = 0
or w^5-1^5= 0
(w-1)(w^4+w^3+w^2+w+1) = 0

by using x^a-y^a series expansion

as w!=1 devide by w-1 to get
w^4+w^3+w^2+w+1 =0
but above is x

so x = 0

Answer 2

THE SIMPLEST SOLUTION

Since w^5 = 1, then w = 1^(1/5), and w has 5 values (the 5 5th roots of 1),
but since 1 is not accepted, then w has 4 values.
This means that w^4 + w³ + w² + w + 1 can have a maximum of 4 values.

We can simplify
x = w^4 + w³ + w² + w + 1 = (w^5 - 1)/(w - 1)

Since w^5 = 1, then
(w^5 - 1)/(w - 1) = (1 - 1)/(w - 1) = 0/(w - 1) = 0

Therefore, the value of w^4 + w³ + w² + w + 1
x = 0
^_^

Answer 3

Hi,

Here's a very simple, sneaky way to do this without long division or anything like that:

look at your equation:

x = w^4 + w^3 + w^2 + w^1 + 1

multiply by w

w*x = w^5 + w^ 4 + w^3 + w^2 + w^1

but w^5 = 1 (as you are told)
so

w*x = 1 + w^4 + w^3 + w^2 + w^1
w*x = x (as the rhs above is what you had in the first place!)

so
w*x=x => w*x - x = 0 => x*(w-1) = 0 => w = 1 or x = 0.. but we are told w is not equal to 1...

I think most of the other respondents are making this too complicated...

Answer 4

x=0. The proofs above for that fact are correct. A more general result is that the sum of the nth roots of 1 is ALWAYS 0. Also, if w is an nth root of 1, then w,w^2, w^3, ..., w^n=1 are all the nth roots of 1. In your example, n=5, but x=0 for any value of n.

Steve

Answer 5

if z^5 =1 = cos (2k.pi + 0) + i sin (2k.pi + 0)
then z is a fifth root of unity.
The first root is unity, let the next be
w = cos (2.pi/5) + i sin (2.pi/5)
you can observe that the other roots are w^2, w^3, and w^4

then, by summing the geometric series
1+ w + w^2 + w^3 + w^4 = (1)(1- w^5) / (1-w)

since w^5 is 1, the numerator becomes zero and x=0

Answer 6

isn't that fact, a million.00000....a million = a million basically like asserting 10 = thirteen? I mean, equivalent ability precisely the comparable. even in spite of the undeniable fact that they are very very close mutually, they're nonetheless 2 different numbers. No evidence could ever substitute that fact. Math is nice good judgment subsequently. additionally on your evidence on the perfect, the line 10x = a million.000000....a million is fake 10x = a million.00000... a million (that being one decimal to the left) in fact, a million.00000... a million isn't a actual quantity. you desire a definitive like a million x 10^-1000 multiply it by 10 and you get 10 x 10^-999 it would be easier to work out in case you mentioned 9x = 9.0000... 9 the completed equation you made relies upon upon that mistake, with the numbers being wisely elevated you get a different consequence.

Answer 7

If w cannot be equal to one, then w must be equal to negative one for at least one of the solutions available. Therefore (-1^4)+(-1^3)+(-1^2)+(-1^1)+1=x will simplify to-- (1)+(-1)+(1)+(-1)+1=x. The ones then mostly cancel out, leaving you with a positive one value for X.

X=+1

Answer 8

w^5 = 1 has five roots, two conjugate complex pairs and 1.
But, you dont want w to be 1.
So,
w = 1/4*5^(1/2)-1/4+1/4*i*2^(1/2)*(5+5^(1/2))^(1/2)
or
w = -1/4*5^(1/2)-1/4+1/4*i*2^(1/2)*(5-5^(1/2))^(1/2)
or
w = -1/4*5^(1/2)-1/4-1/4*i*2^(1/2)*(5-5^(1/2))^(1/2)
or
w = 1/4*5^(1/2)-1/4-1/4*i*2^(1/2)*(5+5^(1/2))^(1/2)

So, in sequence of occurence
w1 =

0.3090 + 0.9511i


w2 =

-0.8090 + 0.5878i


w3 =

-0.8090 - 0.5878i


w4 =

0.3090 - 0.9511i

and again in sequence of occurence of w,
w^4 + w^3 + w^2 + w^1 + 1 = -2.2204e-016 -8.8818e-016i

w^4 + w^3 + w^2 + w^1 + 1 = -2.2204e-016 -1.1102e-016i

w^4 + w^3 + w^2 + w^1 + 1 = -2.2204e-016 +1.1102e-016i

w^4 + w^3 + w^2 + w^1 + 1 = -2.2204e-016 +8.8818e-016i

Thats the answer

Answer 9

w can't equal -1 as -1^5 = -1.

I don't think there is any real value of w if it doesn't equal 1.

Answer 10

It really depends.
Is this a real question for a class or did you make it up?
If it's a real question there needs to be more information
probably provided in your text book or class. For instance,
what group is w an element of? Clearly it repeats the identity
at the fifth power, but it is impossible to determine x unless
the group elements and relations are known.