How do you find all the real cube roots of 8/125?

1 Answer
George C.
Oct 19, 2015

#8/125# has one Real cube root #2/5#

Explanation:

#8 = 2^3# and #125 = 5^3#

So

#root(3)(8/125) = root(3)(2^3/125^3) = root(3)(2^3)/root(3)(5^3) = 2/5#

The other two roots are both Complex:

#2/5 omega = 2/5((-1+i sqrt(3))/2) = (-1+i sqrt(3))/5#

#2/5 omega^2 = 2/5((-1+i sqrt(3))/2) = (-1-i sqrt(3))/5#

where #omega = (-1+i sqrt(3))/2# is the primitive Complex cube root of unity.

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