Assuming you mean the complex roots of the equation:
#x^3=343#
We can find the one real root by taking the third root of both sides:
#root(3)(x^3)=root(3)(343)#
#x=7#
We know that #(x-7)# must be a factor since #x=7# is a root. If we bring everything to one side, we can factor using polynomial long division:
#x^3-343=0#
#(x-7)(x^2+7x+49)=0#
We know when #(x-7)# equals zero, but we can find the remaining roots by solving for when the quadratic factor equals zero. This can be done with the quadratic formula:
#x^2+7x+49=0#
#x=(-7+-sqrt(7^2-4*1*49))/2#
#=>(-7+-sqrt(49-196))/2#
#=>(-7+-sqrt(-147))/2#
#=>(-7+-isqrt(49*3))/2#
#=>(-7+-7sqrt(3)i)/2#
This means that the complex solutions to the equation #x^3-343=0# are
#x=7# and
#x=(-7+-7sqrt(3)i)/2#
