## anonymous one year ago Find the fourth roots of the complex number " z1 = 1 + √3 * i

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1. anonymous

Part I: Write z1 in polar form. Part II: Find the modulus of the roots of z1. Part III: Find the four angles that define the fourth roots of the number z1. Part IV: What are the fourth roots of the equation " z1 = 1 + √3 * i ".

2. anonymous

Is my answer correct? a = 1 b = sqrt(3) sqrt(a^2 + b^2) = sqrt(1 + 3) = sqrt(4) = 2 z = 2 * (1/2 + i * sqrt(3)/2) z = 2 * (cos(pi/3 + 2pi * k) + i * sin(pi/3 + 2pi * k)) z = 2 * (cos((pi/3) * (1 + 6k)) + i * sin((pi/3) * (1 + 6k))) z^(1/4) = 2^(1/4) * (cos((pi/12) * (1 + 6k)) + i * sin((pi/12) * (1 + 6k))) 2^(1/4) * (cos(pi/12) + i * sin(pi/12)) 2^(1/4) * (cos(7pi/12) + i * sin(7pi/12)) 2^(1/4) * (cos(13pi/12) + i * sin(13pi/12)) 2^(1/4) * (cos(19pi/12) + i * sin(19pi/12))

3. anonymous

looks good to me

4. anonymous

modulus is 2

5. anonymous

angle is $$\frac{\pi}{3}$$

6. ganeshie8

you may verify the answers by rising the roots to 4th power you should get back the z1

7. anonymous

and $\frac{\pi}{3}\times \frac{1}{4}=\frac{\pi}{12}$

8. anonymous

all looks swell

9. anonymous

Should I simplify the roots?

10. anonymous

@satellite73

11. ganeshie8

i don't think you can simplify them further

12. anonymous

I meant the angles.

13. ganeshie8

they look good the way they are now

14. ganeshie8

pi/12 what can you simplify here ?

15. anonymous

@satellite73 said it simplifies to pi/3?

16. ganeshie8

Nope, satellite was referring to something else

17. anonymous

ok