Given that x⁸ - 34x⁴ + 1 = 0, x > 0, what is the value of (x³ + x^-3)?

$$\eqalign{ & {x^4}\left( {{x^4} - 34 + \frac{1}{{{x^4}}}} \right) = 0 \cr & {x^4} + \frac{1}{{{x^4}}} = 34 \cr & {x^4} + \frac{1}{{{x^4}}} + 2 = 36 \cr & {x^2} + \frac{1}{{{x^2}}} = 6 \cr & x + \frac{1}{x} = \sqrt 8 \cr & {x^3} + \frac{1}{{{x^3}}} = 8\sqrt 8 - 3\sqrt 8 \cr & {x^3} + \frac{1}{{{x^3}}} = 5\sqrt 8 \cr} $$