anonymous
  • anonymous
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.
Mathematics
schrodinger
  • schrodinger
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amistre64
  • amistre64
4th roots in polar form divide the unit circle into 4 equal parts.... so 90 degree seperations
amistre64
  • amistre64
since the any root of 1 = 1; im assuming they want (1,1) (1,-1) (-1,-1) (-1,1) as answers
anonymous
  • anonymous
isnt it x^4=1?

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amistre64
  • amistre64
I was thinking \(\sqrt[4]{x}\) at \(x=1\)
anonymous
  • anonymous
but i guess it is x=1^(1/4) this is what we call four forth root of 1
anonymous
  • anonymous
x=1,-1,i,-i
amistre64
  • amistre64
90s or the 45s....
amistre64
  • amistre64
if we go with the i stuff; its the 90s and thats prolly the better interpretatio
anonymous
  • anonymous
yup
anonymous
  • anonymous
but for 1 its 0
anonymous
  • anonymous
for -1, pi
amistre64
  • amistre64
there is no pi in the cartesian; just your 1s and 0s for your intercepts
amistre64
  • amistre64
(1,0) (0,1) (-1,0) (0,-1)
amistre64
  • amistre64
but what 'c' is asking for I dunno
amistre64
  • amistre64
maybe \((1-0i)^{1/4}\) ?
amistre64
  • amistre64
or is it simply \(i^4\)
anonymous
  • anonymous
i cant guess any idea
anonymous
  • anonymous
i^4 seems better
amistre64
  • amistre64
im thinking the first since that implies a complex plane and 4 roots
anonymous
  • anonymous
but how (-1,0) and(0,-1)
amistre64
  • amistre64
\((1+0i)^4\) maybe? if forget if its ^4 or ^(1/4) that pops out 4 times
amistre64
  • amistre64
\(sqrt{-6}\) has complex roots right?
anonymous
  • anonymous
look...when we talk about the cube roots of 1 , how we express it? x=1^1/3..no?
amistre64
  • amistre64
i believe so
anonymous
  • anonymous
n for forth root it is x=1^1/4
anonymous
  • anonymous
\[x^4 = 1\]
amistre64
  • amistre64
sqrt(-9) = 3i and we can find both those roots in the complex plane ... gonna have to dbl chk with the wolfram :)
anonymous
  • anonymous
\[(x^2-1)(x^2+1)=0\]
anonymous
  • anonymous
x=1,-1,i,-i
anonymous
  • anonymous
for x=1 r=1 and theta =0
anonymous
  • anonymous
x=-1 r=1, theta =pi -1=cospi
anonymous
  • anonymous
4th roots of 1 are 1, -1, i, -1
anonymous
  • 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
  • anonymous
did u get part c?
anonymous
  • anonymous
what is part c?
anonymous
  • anonymous
read the post:P
anonymous
  • anonymous
oh yes they are all powers of i
anonymous
  • anonymous
i, i^2, i^3, i^4 finito
anonymous
  • anonymous
no question says" Show how they can be expressed as powers of one fixed fourth root of 1."
anonymous
  • anonymous
that fixed root is i.
anonymous
  • anonymous
you cannot express i as a power of 1. i assume they mean integral powers

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