1. **State the problem:** We have a cuboid with volume $2$ m³ and three sides: $5 - \sqrt{3}$ cm, $2 + \sqrt{3}$ cm, and $x$ cm (the missing side). We need to find $x$ in simplified surd form with a rationalised denominator. 2. **Convert units:** Since volume is in m³ and sides are in cm, convert sides to meters: $$5 - \sqrt{3} \text{ cm} = \frac{5 - \sqrt{3}}{100} \text{ m}, \quad 2 + \sqrt{3} \text{ cm} = \frac{2 + \sqrt{3}}{100} \text{ m}$$ 3. **Volume formula:** Volume of cuboid is product of sides: $$V = \text{side}_1 \times \text{side}_2 \times \text{side}_3$$ 4. **Set up equation:** $$2 = \left(\frac{5 - \sqrt{3}}{100}\right) \times \left(\frac{2 + \sqrt{3}}{100}\right) \times x$$ 5. **Simplify product of known sides:** $$\left(5 - \sqrt{3}\right) \times \left(2 + \sqrt{3}\right) = 5 \times 2 + 5 \times \sqrt{3} - 2 \times \sqrt{3} - \sqrt{3} \times \sqrt{3} = 10 + 5\sqrt{3} - 2\sqrt{3} - 3 = 7 + 3\sqrt{3}$$ 6. **Substitute back:** $$2 = \frac{7 + 3\sqrt{3}}{10000} \times x$$ 7. **Solve for $x$:** $$x = \frac{2 \times 10000}{7 + 3\sqrt{3}} = \frac{20000}{7 + 3\sqrt{3}}$$ 8. **Rationalise denominator:** Multiply numerator and denominator by conjugate $7 - 3\sqrt{3}$: $$x = \frac{20000 (7 - 3\sqrt{3})}{(7 + 3\sqrt{3})(7 - 3\sqrt{3})}$$ 9. **Calculate denominator:** $$(7)^2 - (3\sqrt{3})^2 = 49 - 27 = 22$$ 10. **Simplify $x$:** $$x = \frac{20000 (7 - 3\sqrt{3})}{22} = \frac{20000}{22} (7 - 3\sqrt{3}) = \frac{10000}{11} (7 - 3\sqrt{3})$$ 11. **Final answer:** $$\boxed{x = \frac{10000}{11} (7 - 3\sqrt{3}) \text{ meters}}$$ Since original sides were in cm, convert $x$ back to cm: $$x = \frac{10000}{11} (7 - 3\sqrt{3}) \text{ m} = \frac{10000}{11} (7 - 3\sqrt{3}) \times 100 \text{ cm} = \frac{1000000}{11} (7 - 3\sqrt{3}) \text{ cm}$$ This is a very large number, so better to keep $x$ in meters or re-check units if needed. But the problem states volume in m³ and sides in cm, so this is consistent. **Summary:** The missing side $x$ in meters is $$x = \frac{10000}{11} (7 - 3\sqrt{3})$$