## paulzord 4 years ago 'complex' numbers? Suppose w6 = -64i. Determine the six values of w.

1. paulzord

hm i have no idea how to even start this...

2. mathgirl

is this supposed to be w raised to the 6th power?

3. paulzord

ah yes it is sorry

4. satellite73

is this $w^6=-64i$?

5. satellite73

if so this is not so bad.

6. paulzord

yes it is $w ^{6} = -64i$

7. mathgirl

you just take the 6th root, right sat?

8. satellite73

hello math girl. this site is very slow for me tonight

9. mathgirl

figure out what number you multiply by itself 6 times

10. paulzord

2

11. mathgirl

that gives you 64, and then stick on i on it

12. satellite73

yeah just take the 6th roots. if working in degrees write $-64i=64(\cos(270)+i\sin(270))$ and then take the 6ht root of 64 which is 2 and divide 270 by 6 to get 45 so first answer is $2(\cos(45)+i\sin(45))$

13. mathgirl

where'd the trig stuff come from?

14. satellite73

degrees means it is easier for me to divide and get whole number not write fractions in latex

15. mathgirl

i must be confused about the type of problem here

16. paulzord

how do yo know its 270?

17. satellite73

think of where -64i is in the complex plane. it is straight down 64 units, so trig form is what i wrote. you can also use $-64i=64e^{\frac{3\pi}{2}i}$ and do it that way

18. satellite73

oh how to i know it is 270?

19. satellite73

draw the complex plane and imagine where -64i is. it is straight down 64 units, think of it as $0-64i$

20. mathgirl

what math class is this? i never learned this stuff

21. satellite73

you can do this as soon as you learn trig

22. satellite73

write $a+bi$ as $r(\cos(\theta)+i\sin(\theta)$ where $r=\sqrt{a^2+b^2}$ and $\tan(\frac{b}{a})=\theta$

23. satellite73

in any case when you want the sixth roots, take the sixth root of the absolute value and divide the angle by 6

24. abtrehearn

If$w^{6} = -64 i,$then$|w| = \sqrt[6]{64}= 2.$The angle $\theta = 3 \pi/2,$so one of the sixth roots of -64i will be a vertex of the regular hexagon lf side length 2 centered at z = 0, and making an angle $\theta/6 = (3 \pi/2)/6 = \pi/4.$ That vertex is at$z = 2 \cos(\pi/4) + 2i \sin(\pi/4) = \sqrt{2} + i \sqrt{2}.$All the sixth roots are$2 \cos(\pi/4 + k \pi/3) + 2i \sin(\pi/4 + k \pi / 3),$where k is an integer.

25. satellite73

so first answer is $2(\cos(45)+i\sin(45)$ $=2(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)$ $=\sqrt{2}+\sqrt{2}i$

26. satellite73

to get second answer just divide circle up into 6 equal parts with one part at 45 degrees. or go around the circle again and write $-64i=64(\cos(630)+i\sin(630))$ and now divide 630 by 6 to get 105 so second answer is $2(\cos(105)+i\sin(105))$ etc

27. satellite73

28. paulzord

how did you get that angle 3pi/2 when it looks like im meant to go tan 0/2

29. satellite73

here is a picture of the unit circle in the complex plane. you can see that the angle associated with i is $\frac{3\pi }{2}$

30. satellite73

and by the way it is $\theta =\tan(\frac{b}{a})$ which is this case is $\tan(\frac{-64}{0})$ which is undefined

31. satellite73
32. abtrehearn

The angle$3 \pi/2$comes from plotting (0, -64) on the x-y plane and measuring counterclockwise from the positive x-axis to the point. That's three right angles, or $3 \pi/2$radians.

33. paulzord

oh... click

34. paulzord

no wonder the 6th root of it is pi/2?

35. satellite73

if you want to do this the slow donkey way, which may not be a bad idea the first time you see it, start with 270 degrees (i know we really should work in radians but this will make my typing easier) then divide 270 by 6 for the next one add 360 to 270 to get 630, divide by 6 to get 105

36. satellite73

then for the next one add 360 again to get 990 and divide by 6 to get 165 etc

37. satellite73

around again you get 990+360=1350 divide by 6 to get 225 around again you get 1350+360=1710 divide by 6 get 285 around again you get 1710+360=2070 divide by 6 get 345 and finally around again you get 2070+360 = 2430 divide by 6 get 405 which is the same as 45 so you are back where you started

38. satellite73

this is certainly not the elegant way to do it, but it gets the point across

39. abtrehearn

45 + 60 = 105 105 + 60 = 165 165 + 60 = 225 225 + 60 = 285 285 + 60 = 345. Those are the angles for the vertices in degrees.