Question

a) Find all fourth roots of 1 in polar form.
b) Express them in Cartesian form.
c) Show how they can be expressed as powers of one fixed fourth root of 1.

Answer 1

4th roots in polar form divide the unit circle into 4 equal parts.... so 90 degree seperations

Answer 2

since the any root of 1 = 1; im assuming they want (1,1) (1,-1) (-1,-1) (-1,1) as answers

Answer 3

isnt it x^4=1?

Answer 4

I was thinking \(\sqrt[4]{x}\) at \(x=1\)

Answer 5

but i guess it is x=1^(1/4)
this is what we call four forth root of 1

Answer 6

x=1,-1,i,-i

Answer 7

90s or the 45s....

Answer 8

if we go with the i stuff; its the 90s and thats prolly the better interpretatio

Answer 9

yup

Answer 10

but for 1 its 0

Answer 11

for -1, pi

Answer 12

there is no pi in the cartesian; just your 1s and 0s for your intercepts

Answer 13

(1,0)
(0,1)
(-1,0)
(0,-1)

Answer 14

but what 'c' is asking for I dunno

Answer 15

maybe \((1-0i)^{1/4}\) ?

Answer 16

or is it simply \(i^4\)

Answer 17

i cant guess any idea

Answer 18

i^4 seems better

Answer 19

im thinking the first since that implies a complex plane and 4 roots

Answer 20

but how (-1,0) and(0,-1)

Answer 21

\((1+0i)^4\) maybe? if forget if its ^4 or ^(1/4) that pops out 4 times

Answer 22

\(sqrt{-6}\) has complex roots right?

Answer 23

look...when we talk about the cube roots of 1 , how we express it?
x=1^1/3..no?

Answer 24

i believe so

Answer 25

n for forth root it is
x=1^1/4

Answer 26

\[x^4 = 1\]

Answer 27

sqrt(-9) = 3i and we can find both those roots in the complex plane ... gonna have to dbl chk with the wolfram :)

Answer 28

\[(x^2-1)(x^2+1)=0\]

Answer 29

x=1,-1,i,-i

Answer 30

for x=1

r=1 and theta =0

Answer 31

x=-1
r=1, theta =pi
-1=cospi

Answer 32

4th roots of 1 are 1, -1, i, -1

Answer 33

you know one answer is 1. divide unit circle (in complex plane) into 4 equal parts and you will see i, -1, -i

Answer 34

did u get part c?

Answer 35

what is part c?

Answer 36

read the post:P

Answer 37

oh yes they are all powers of i

Answer 38

i, i^2, i^3, i^4 finito

Answer 39

no question says" Show how they can be expressed as powers of one fixed fourth root of 1."

Answer 40

that fixed root is i.

Answer 41

you cannot express i as a power of 1. i assume they mean integral powers