Find the fourth roots of #-17i#?

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Answer 1 · 2016-05-23T13:37:47

Fourth roots of #-17i# are #root(4)17e^((3pi)/8i)#, #root(4)17e^((7pi)/8i)#, #root(4)17e^((11pi)/8i)# and #root(4)17e^((15pi)/8i)#

Let us wrote #-17i# in polar form and as its modulus is #17# and number is #117xx(-i)#, we can write it as

#-17i=17e^((3pi)/2i)#

Let #x=root(4)(-17i)#, then

#x^4=-17i=17e^((3pi)/2i)#

Hence #x=root(4)17e^((3pi)/8i)#

Other roots can be obtained by adding #pi/2# to #(3pi)/8# in succession (as they are one-fourth of #2pi#).

i.e. #root(4)17e^((7pi)/8i)#, #root(4)17e^((11pi)/8i)# and #root(4)17e^((15pi)/8i)#

To find them in the form of #a+bi#, you can use the following

#re^(itheta)=rcostheta+isintheta#