1. **Stating the problem:** We are given a cuboid with dimensions $5 - \sqrt{3}$ cm, $2 + \sqrt{3}$ cm, and an unknown side $x$ cm. The volume is $2$ m³. We need to find $x$ in simplified surd form with a rationalised denominator. 2. **Convert volume units:** Since dimensions are in cm and volume is in m³, convert $2$ m³ to cm³: $$2 \text{ m}^3 = 2 \times 100^3 = 2 \times 1,000,000 = 2,000,000 \text{ cm}^3$$ 3. **Volume formula:** Volume $V = \text{length} \times \text{width} \times \text{height}$ $$2,000,000 = (5 - \sqrt{3})(2 + \sqrt{3})x$$ 4. **Multiply the known sides:** $$(5 - \sqrt{3})(2 + \sqrt{3}) = 5 \times 2 + 5 \times \sqrt{3} - 2 \times \sqrt{3} - \sqrt{3} \times \sqrt{3}$$ $$= 10 + 5\sqrt{3} - 2\sqrt{3} - 3 = (10 - 3) + (5\sqrt{3} - 2\sqrt{3}) = 7 + 3\sqrt{3}$$ 5. **Solve for $x$:** $$x = \frac{2,000,000}{7 + 3\sqrt{3}}$$ 6. **Rationalise the denominator:** Multiply numerator and denominator by the conjugate $7 - 3\sqrt{3}$: $$x = \frac{2,000,000 (7 - 3\sqrt{3})}{(7 + 3\sqrt{3})(7 - 3\sqrt{3})}$$ 7. **Calculate denominator:** $$(7)^2 - (3\sqrt{3})^2 = 49 - 9 \times 3 = 49 - 27 = 22$$ 8. **Final expression for $x$:** $$x = \frac{2,000,000 (7 - 3\sqrt{3})}{22} = \frac{2,000,000}{22} (7 - 3\sqrt{3})$$ 9. **Simplify the fraction:** $$\frac{2,000,000}{22} = \frac{1,000,000}{11}$$ 10. **Write final answer:** $$x = \frac{1,000,000}{11} (7 - 3\sqrt{3})$$ This is the value of the missing side in fully simplified surd form with a rationalised denominator.