SOLUTION: (a) Find the sixth roots of 1.
(b) Find the fifth roots of 32.
Algebra.Com
Question 1183762: (a) Find the sixth roots of 1.
(b) Find the fifth roots of 32.
Found 2 solutions by Boreal, greenestamps:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
the first ia -1 and 1 if you want the sixth root. If you want the six rootS, they are -1 with multiplicity 3 and 1 with multiplicity 3. I think you want -1, 1 with this question, but the wording matters.
the second is 2.
Answer by greenestamps(13206) (Show Source): You can put this solution on YOUR website!
The answer from the other tutor is incomplete/incorrect. There are 6 6th roots of 1 (or any other number); there are 5 5th roots of 32 (or any other number). The response from the other tutor only shows the real roots -- 2 of the 6 6th roots of 1, and 1 of the 5 5th roots of 32.
De Moivre's Theorem solves this kind of problem easily.
Given a number represented in trigonometric form
or ,
de Moivre's Theorem says that the "primary" n-th root is
and the other n-th roots have the same magnitude as the primary root and are distributed around the Argand plane at intervals of
(a) The 6 6th roots of 1
The primary root is
The other 6th roots of 1 are distributed about the Argand plane at intervals of ; so the 6 6th roots of 1 are
cis(0) = 1
cis(pi/3) = 1/2+i*sqrt(3)/2
cis(2pi/3) = -1/2+i*sqrt(3)/2
cis(pi) = -1
cis(4pi/3) = -1/2-i*sqrt(3)/2
cis(5pi/3) = 1/2-i*sqrt(3)/2
(b) The 5 5th roots of 32
The primary root is
The other 5th roots of 1 are distributed about the Argand plane at intervals of 2*cis(pi/5); so the 5 5th roots of 32 are:
2*cis(0) = 2
2*cis(pi/5)
2*cis(2pi/5)
2*cis(3pi/5)
2*cis(4pi/5)
RELATED QUESTIONS
find the fifth roots of i (answered by Fombitz)
Find the fifth roots of... (answered by ikleyn)
Find all the sixth roots of... (answered by CharlesG2,JBarnum)
Find all the sixth roots of... (answered by CharlesG2)
Find all the sixth roots of... (answered by robertb)
Find the fifth roots of 32(cos 280° + i sin... (answered by srinivas.g)
Find all the fifth roots of... (answered by cininnatus)
1.If α, b, g are the roots of the equation
x3─7x2 + x + 5 = 0
Find (answered by khwang)
find the square roots of 1+6i
(answered by dkppathak,ikleyn)