1. **State the problem:** Given the equation $x^2 = 623 - \frac{1}{x^3}$, find the value of $x^3 + \frac{1}{x^3}$.\n\n2. **Rewrite the given equation:** Multiply both sides by $x^3$ to clear the denominator:\n$$x^2 \cdot x^3 = x^3 \cdot 623 - x^3 \cdot \frac{1}{x^3}$$\nwhich simplifies to\n$$x^5 = 623x^3 - 1$$\n\n3. **Express $x^5$ in terms of $x^3$:** From the above, we have\n$$x^5 - 623x^3 + 1 = 0$$\n\n4. **Introduce substitution:** Let $y = x^3 + \frac{1}{x^3}$. Our goal is to find $y$.\n\n5. **Relate $y$ to $x$:** Note that\n$$\left(x + \frac{1}{x}\right)^3 = x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} = x^3 + \frac{1}{x^3} + 3\left(x + \frac{1}{x}\right)$$\nSo,\n$$y = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right)$$\n\n6. **Find $x + \frac{1}{x}$:** From the original equation, rewrite as\n$$x^2 = 623 - \frac{1}{x^3}$$\nMultiply both sides by $x$:\n$$x^3 = 623x - \frac{1}{x^2}$$\nBut this is complicated; instead, try to find $x + \frac{1}{x}$ by other means.\n\n7. **Try to find $x + \frac{1}{x}$ using the given:** Multiply the original equation by $x^3$:\n$$x^5 = 623x^3 - 1$$\nRewrite as\n$$x^5 - 623x^3 + 1 = 0$$\n\n8. **Let $z = x^3$, then the equation becomes:**\n$$z^2 - 623z + 1 = 0$$\nSolve for $z$:\n$$z = \frac{623 \pm \sqrt{623^2 - 4}}{2}$$\n\n9. **Calculate the discriminant:**\n$$623^2 - 4 = 388129 - 4 = 388125$$\n\n10. **Approximate $\sqrt{388125}$:**\n$$\sqrt{388125} \approx 622.99$$\n\n11. **Find roots:**\n$$z_1 = \frac{623 + 622.99}{2} = \frac{1245.99}{2} = 622.995$$\n$$z_2 = \frac{623 - 622.99}{2} = \frac{0.01}{2} = 0.005$$\n\n12. **Calculate $z + \frac{1}{z}$:**\nFor $z_1$:\n$$622.995 + \frac{1}{622.995} \approx 622.995 + 0.001605 = 623.9966$$\nFor $z_2$:\n$$0.005 + \frac{1}{0.005} = 0.005 + 200 = 200.005$$\n\n13. **Since $z = x^3$, $y = x^3 + \frac{1}{x^3} = z + \frac{1}{z}$. So the possible values are approximately $623.9966$ or $200.005$.\n\n14. **Final answer:**\n$$x^3 + \frac{1}{x^3} \approx 624 \text{ or } 200$$