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If the six roots of x^6 = -64 are writt...

If the six roots of `x^6 = -64` are written in the form `a + ib`, where a and b are real, then the product ofthose roots for which `a lt 0` is

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The correct Answer is:
5
`x^(6) = - 64`
`rArr ((x)/(2i))^(6) =1`
`rArr (x)/(2i) = cos((2npi)/(6)) +isin((2npi)/(6)),` Where n = 0,1,2,3,4,5
`rArr x = 2i cos ((npi)/(3)) -2sin((npi)/(2))`, where n =0,1,2,3,4,5
For n=4 and 5 we have positive real part. Hence, the requrired product is
`[2icos ((4pi)/(3)) -2sin((4pi)/(3))][2cosi((5pi)/(3)) - 2sin((5pi)/(3))]`
`=4[-i(1)/(2)+(sqrt(3))/(2)][i(1)/(2)+(sqrt(3))/(2)]`
`= 4`
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