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