Find the all value of (-1)^1/6?

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Answer 1 · 2018-02-16T09:34:47

#root(6)(-1)#

#(-1)^(1/6)#

Know that, #a^(b/c)=root(c)(a^b)#

So, the original expression becomes

#root(6)((-1)^1)#

#=root(6)(-1)#

From here, it doesn't look like you can simplify this further, or maybe you can, but you would get a really weird complex number...

Answer 2 · 2018-02-16T09:40:47

#=> z = e^( (ipi)/6 ( 1+2k) ) #

#k = { 0,1,2,3,4,5 } #

If you understand complex numbers...

Let #z = (-1)^(1/6) #

#=> z^6 = -1 #

Hence this is the equation we can try to solve to find the value of #z#

#-1 = e^(ipi ) #

#=> z^6 = e^(ipi)#

We know #e^(2kipi ) =1#

#AA k in ZZ #

Multiply by this, is just the same as multiplying by 1:

#=> z^6 = e^(ipi) * e^(2kipi) #

#=> z^6 = e^(ipi (1+2k) ) #

#=> z = e^( (ipi)/6 ( 1+2k) ) , k = { 0,1,2,3,4,5 } #

As all the other #k# yields repeated roots...