1. **State the problem:** We have a trapezium with area $6\sqrt{21}$ cm², a top side of 5 cm, a height of $\sqrt{7}$ cm, and a base labeled $f$ cm. We need to find $f$ in the form $x\sqrt{y} - z$ where $x,y,z$ are positive integers. 2. **Formula for the area of a trapezium:** $$\text{Area} = \frac{(\text{base}_1 + \text{base}_2)}{2} \times \text{height}$$ Here, base$_1 = f$, base$_2 = 5$, height $= \sqrt{7}$. 3. **Set up the equation:** $$6\sqrt{21} = \frac{f + 5}{2} \times \sqrt{7}$$ 4. **Isolate $f + 5$:** Multiply both sides by 2: $$2 \times 6\sqrt{21} = (f + 5) \times \sqrt{7}$$ $$12\sqrt{21} = (f + 5) \sqrt{7}$$ 5. **Divide both sides by $\sqrt{7}$:** $$\frac{12\sqrt{21}}{\sqrt{7}} = f + 5$$ 6. **Simplify the fraction:** Recall $\frac{\sqrt{21}}{\sqrt{7}} = \sqrt{\frac{21}{7}} = \sqrt{3}$. So, $$12 \sqrt{3} = f + 5$$ 7. **Solve for $f$:** $$f = 12 \sqrt{3} - 5$$ 8. **Identify $x$, $y$, and $z$:** Comparing to $x\sqrt{y} - z$, we have: $$x = 12, \quad y = 3, \quad z = 5$$ **Final answer:** $$f = 12 \sqrt{3} - 5$$