## anonymous 5 years ago 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.

1. amistre64

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

2. amistre64

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

3. anonymous

isnt it x^4=1?

4. amistre64

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

5. anonymous

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

6. anonymous

x=1,-1,i,-i

7. amistre64

90s or the 45s....

8. amistre64

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

9. anonymous

yup

10. anonymous

but for 1 its 0

11. anonymous

for -1, pi

12. amistre64

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

13. amistre64

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

14. amistre64

but what 'c' is asking for I dunno

15. amistre64

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

16. amistre64

or is it simply $$i^4$$

17. anonymous

i cant guess any idea

18. anonymous

i^4 seems better

19. amistre64

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

20. anonymous

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

21. amistre64

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

22. amistre64

$$sqrt{-6}$$ has complex roots right?

23. anonymous

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

24. amistre64

i believe so

25. anonymous

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

26. anonymous

$x^4 = 1$

27. amistre64

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

28. anonymous

$(x^2-1)(x^2+1)=0$

29. anonymous

x=1,-1,i,-i

30. anonymous

for x=1 r=1 and theta =0

31. anonymous

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

32. anonymous

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

33. anonymous

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

34. anonymous

did u get part c?

35. anonymous

what is part c?

36. anonymous

37. anonymous

oh yes they are all powers of i

38. anonymous

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

39. anonymous

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

40. anonymous

that fixed root is i.

41. anonymous

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