Question

Can someone help me find the seventh roots of j

Answer 1

is that the complete question?

Answer 2

Yes

Find the seventh roots of j (Remember j is equivalent to 0+1j)

that is all it says

Answer 3

perhaps it is what we usually call \(i\) aka \(\sqrt{-1}\) is that possible?

Answer 4

Yeah 'j' in electrical engineering is same as 'i' in math

Answer 5

yes instead of "i" we use "j"

Answer 6

oooh
well this will be no joy, but totally doable'
do you know how to write \(j\) in trig form or exponential form?

Answer 7

yes im familiar but i am mostly using polar form, exponential form and rectangular form

Answer 8

then write it in polar form, divide the angle by 7

Answer 9

then add \(2\pi\) to the angle, divide by 7 again
do this seven times

Answer 10

but you cant divide by 0, in order to find the angle you have to do inverse tangent right?

Answer 11

\[j = 1\angle 90 = x^7\]

Your goal here is to find all x that satisfy above equation

Answer 12

oh no no no you do not use inverse tangent (although you can)
just look at a picture

Answer 13

Remember this rule ?
\[(r\angle \theta )^n = r^n \angle n\theta\]

Answer 14

no
what does it mean/

Answer 15

It is same as demoivre thm in polar form

Answer 16

oh

Answer 17

oh ok

Answer 18

and do i fill in "K" using the number 1-6

Answer 19

no idea what K is but the first root would be \[\cos(\frac{\pi}{14})+j\sin(\frac{\pi}{14})\]

Answer 20

Can you find one root by staring at
\[x^7 = 1\angle 90\]
?

Answer 21

since the picture makes it pretty clear that \(\theta=\frac{\pi}{2}\) and when you divide by 7 you get \(\frac{\pi}{14}\)

Answer 22

\[1e ^{j1.57}\]

Answer 23

How ?

Answer 24

changed it to radian

Answer 25

you writing in exponential from it is the same just \[\huge e^{\frac{\pi}{14}i}\]

Answer 26

Could you keep it in fraction form please as I'm on mobile and can't know whatvyiuve done to get 1.57 ?

Answer 27

to get 1.57 i multiplied 90 by pi then divided by 180

Answer 28

shouldn't i find the seventh root of 1

Answer 29

forget fractions, the answers are irrational

Answer 30

you mean the real seventh root of 1????
yes you need it , but it is 1 for sure

Answer 31

You could leave the angle as pi/2 instead of writing in decimal form 1.57

Answer 32

yeah ok i think i got it

Answer 33

is dividing 360 by 7 required

Answer 34

Have you found one root by staring at the previous equation ?

Answer 35

0+1j ?

Answer 36

\[x^7=1\angle 90\]

Let the required root x in polar form be \(r\angle \theta\)
\[(r\angle \theta)^7=1\angle 90\]
Using demoivres thm this becomes
\[r^7\angle 7\theta=1\angle 90\]

Answer 37

Comparing the magnitude and angle both sides
\[r^7=1\]
\[7\theta = 90\]

Answer 38

Oh sorry bout that it was required to be written in rectangular form

Answer 39

You can convert it back to rectangular form in the end. Multiplication and exponenets are easier to work in polar form

Answer 40

i get you now all thats left is for me to find the other 5, thanks alots

Answer 41

Okay good. So what is one root from above eqns ?

Answer 42

180

Answer 43

No

Answer 44

Its easy if you understand what exactly I'm asking

Answer 45

arnt they suppose to have the same size angles between each others

Answer 46

\[r^7 = 1\]
\[7\theta = 90\]

Answer 47

oh the same size between 7 angles

Answer 48

Can you simply solve \(r\) and \(\theta\) from above two equations ?

Answer 49

r=1

\[\theta \approx12.86\]

Answer 50

Yes, but it looks nice if you leave it as a fraction 90/7

Answer 51

ok and i am adding this number

Answer 52

Maybe first convince yourself that \(1\angle 90/7\) is indeed a 7th root of j

Answer 53

it is similar to another equation i did finding the third roots