Find all roots of the equation z^5=i, i.e. find the five values of i^1/5 and show...

Find all roots of the equation z^5=i, i.e. find the five values of i^1/5 and show them on an Argand diagram. Show all working

Expert Solution

zⁿ = rⁿ(cos n + i sin n), where n = power, r = modulus, = angle of radius.

Here n = 1/5, r = 1, ? = /2, z=i^1/5

i^1/5 = [ 1^1/5 (cos + i sin)^1/5 ]

i^1/5 {(cis /2)^1/5 , (cis 3/2)^1/5 , (cis 5/2)^1/5 , (cis 7/2)^1/5 ,(cis 9/2)^1/5 }

i^1/5 {(cis /10), (cis 3/10), (cis 5/10), (cis 7/10), (cis 9/10) }

Note: all of these values are irrational. Also, symbol "cis ": "cis" is a shorthand for cos + i sin

Argand diagram:

milcah answered 1 year ago

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