1. **State the problem:** We are given a trapezium with area $6\sqrt{21}$ cm², top base $5$ cm, height $\sqrt{7}$ cm, and bottom base $f$ cm. We need to find $f$ in the form $x\sqrt{y} - z$ where $x,y,z$ are positive integers. 2. **Formula for area of trapezium:** $$\text{Area} = \frac{(\text{top base} + \text{bottom base}) \times \text{height}}{2}$$ 3. **Substitute known values:** $$6\sqrt{21} = \frac{(5 + f) \times \sqrt{7}}{2}$$ 4. **Multiply both sides by 2:** $$2 \times 6\sqrt{21} = (5 + f) \times \sqrt{7}$$ $$12\sqrt{21} = (5 + f) \times \sqrt{7}$$ 5. **Divide both sides by $\sqrt{7}$:** $$\frac{12\sqrt{21}}{\sqrt{7}} = 5 + f$$ 6. **Simplify the left side:** Since $\sqrt{21} = \sqrt{7 \times 3} = \sqrt{7} \times \sqrt{3}$, $$\frac{12 \times \sqrt{7} \times \sqrt{3}}{\sqrt{7}} = 12 \sqrt{3}$$ 7. **So:** $$12 \sqrt{3} = 5 + f$$ 8. **Solve for $f$:** $$f = 12 \sqrt{3} - 5$$ 9. **Identify $x,y,z$:** $x = 12$, $y = 3$, $z = 5$ **Final answer:** $$f = 12 \sqrt{3} - 5$$