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.

- anonymous

- schrodinger

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- amistre64

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

- amistre64

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

- anonymous

isnt it x^4=1?

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## More answers

- amistre64

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

- anonymous

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

- anonymous

x=1,-1,i,-i

- amistre64

90s or the 45s....

- amistre64

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

- anonymous

yup

- anonymous

but for 1 its 0

- anonymous

for -1, pi

- amistre64

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

- amistre64

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

- amistre64

but what 'c' is asking for I dunno

- amistre64

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

- amistre64

or is it simply \(i^4\)

- anonymous

i cant guess any idea

- anonymous

i^4 seems better

- amistre64

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

- anonymous

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

- amistre64

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

- amistre64

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

- anonymous

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

- amistre64

i believe so

- anonymous

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

- anonymous

\[x^4 = 1\]

- amistre64

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

- anonymous

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

- anonymous

x=1,-1,i,-i

- anonymous

for x=1
r=1 and theta =0

- anonymous

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

- anonymous

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

- anonymous

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

- anonymous

did u get part c?

- anonymous

what is part c?

- anonymous

read the post:P

- anonymous

oh yes they are all powers of i

- anonymous

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

- anonymous

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

- anonymous

that fixed root is i.

- anonymous

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

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