Here's my stab.

I took your equation and expanded it to;

(x^6 + 30x - 1) / x^3 = 0 ( I don't feel like typing out the steps, I'm sure you can do it too.)

So this only equals zero if;

x^6 + 30x - 1 = 0

Let x^3 = y, then;

y^2 + 30y - 1 = 0

Using the quadratic equation gives;

y = .033296378.. and -30.03329638...

Since y = x^3, x = y^1/3

x = .321710818... and -3.1081632...

edit*

I checked this with my TI-89 and my answer was confirmed correct. The slight error from being exact is only from rounding the value of x.

The precise answer is;

-(√(226) + 15)^(1/3) and (√(226) - 15)^(1/3)

Which agrees with what I had earlier.

We don' need no stinkin' calculators....