Question

10. find the 4th roots of the complex number z1=1+sqrt3+i

part1: write z1 in polar form
part2: find the mudulus of the root of z1
part3: find the four angles that define the 4th roots of the number z1
part4: what are the fourth roots of z1=sqrt3+i

Answer 1

\[z ^{4}=(1+\sqrt{3})+i\] is this the correct expression ?

Answer 2

its acually z with a 1 at the bottom, and everything else is correct

Answer 3

\[z _{1}=2.9<69.67^{o}\]

Answer 4

thats in polar form?

Answer 5

yep

Answer 6

you know how to find the muduus

Answer 7

yh, the modulus is just adding and squaring the x and y components to find the square root of the two

Answer 8

the \[|z _{1}|=2.9\]

Answer 9

ok part3 and 4?

Answer 10

yh, what's the power of z ? it should be 4 ryt, since they need four roots ?

Answer 11

and part4

Answer 12

1.3<17.42, 1.3<107.42, 1.3<197.42 and 1.3<287.42

Answer 13

the correct angles are 17.42,107.42, 197.42 and 287.42

Answer 14

part 4.
\[z _{1}=\sqrt{3}+i\]

Answer 15

\[z _{1}= 2<30^{o}\]
fourth roots in polar form
0.5<7.5, 0.5<97.5, 0.5<187.5 and 0.5<277.5

Answer 16

to convert to rectangular form, use de Moivre's theorem,
\[r=[\cos(\theta)+isin(\theta)]\]