SOLUTION: how do you find the nth term of a negative complex number? for example the fourth roots of -4

Question 29843: how do you find the nth term of a negative complex number?
for example the fourth roots of -4
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
how do you find the nth term of a negative complex number?
I HOPE YOU MEAN N th. ROOT NOT AND NOT N th. TERM..OF A COMPLEX NUMBER ..CAN BE POSITIVE OR NEGATIVE...
LET THE COMPLEX NUMBER BE Z = A+Bi IN GENERAL WHICH CAN ALWAYS BE WRITTEN AS
Z = R*(COS X + i SIN X ) WHERE R = SQRT(A^2+B^2) AND COS X = A/(SQRT(A^2+B^2)
AND SIN X = B/(SQRT(A^2+B^2)
SINCE N th.ROOT OF R IS EASY TO FIND (R^1/N),R BEING POSITIVE, WE NEED TO ONLY FIND Nth.ROOT OF SAY Z = COS X +i SIN X
Z^1/N = (COS X +i SIN X)^1/N...AS PER DEMOVIERS THEOREM THIS IS EQUAL TO
= COS (X/N) +i SIN (X/N)..SO NOW WE USE THE GENERAL FORMULA FOR TRIGNOMETRIC EQNS ..THAT IS ..IF COS X = P SAY THEN X = COS ^-1(P)..ITS GENERAL SOLUTION IS
X=2K*(PI)+COS^-1(P)...GIVE K=1,2,3 ETC.. TO GET ALL THE DIFFERENT ROOTS.
for example the fourth roots of -4
Z = -4 =4*(-1)...SINCE 4 TH. ROOT OF 4 CAN BE EASILY FOUND OUT .LET US FIND FOURTH ROOTS OF -1.WE CAN MULTIPLY THAT ANSWER WITH 4TH ROOT OF +4 TO GET THE TOTAL ANSWER.
Z = COS (0) +i SIN (0)..
= COS (2K*PI+0)+iSIN (2K*PI+0)=COS (2K*PI)+i SIN (2K*PI)
4TH.ROOTS OF THIS ARE AS PER ABOVE FORMULA...
Z^1/4 =COS (2K*PI)/4+i SIN (2K*PI)/4...PUTTING K=1,2,3 AND 4 TO GET THE 4 ROOTS WE GET...
COS (2*1*PI)/4+i SIN (2*1*PI)/4...,
COS (2*2*PI)/4+i SIN (2*2*PI)/4...,
COS (2*3*PI)/4+i SIN (2*3*PI)/4...,
COS (2*4*PI)/4+i SIN (2*4*PI)/4...AS THE 4 ROOTS.

RELATED QUESTIONS
Please tell how to graphically locate the nth roots of a complex number, given the... (answered by KMST)

Do the sum of the n nth roots of a complex number z always equals zero? Show this using... (answered by ikleyn,Edwin McCravy)

How do you find the 100th and nth term of a... (answered by stanbon,solver91311)

how do you find the number of complex roots in a equation such as {{{ a^2-81=0... (answered by richard1234)

For x^3-x^2+7x-7=0 a)how do you find the number of complex roots? b)how do you find the (answered by lwsshak3)

How could you use Descartes' rule and the Fundamental Theorem of Algebra to predict the... (answered by rothauserc)

how do u find the 10th term and the nth term of a sequence (answered by fractalier,ikleyn)

What are the complex roots of f(x)=x^4+x^2+4 and how do you find... (answered by stanbon)

How do you find the nth term of the geometric sequence whose initial term and common... (answered by ikleyn)