1. **Problem 2: Find the area and perimeter of the rectangle with sides $\sqrt{45}$ cm and $\sqrt{80}$ cm.** 2. **Formula for area and perimeter of a rectangle:** - Area $= \text{length} \times \text{width}$ - Perimeter $= 2(\text{length} + \text{width})$ 3. **Calculate the area:** $$\text{Area} = \sqrt{45} \times \sqrt{80} = \sqrt{45 \times 80} = \sqrt{3600} = 60$$ 4. **Calculate the perimeter:** $$\text{Perimeter} = 2(\sqrt{45} + \sqrt{80})$$ Simplify each surd: $$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$ $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ So, $$\text{Perimeter} = 2(3\sqrt{5} + 4\sqrt{5}) = 2(7\sqrt{5}) = 14\sqrt{5}$$ --- 5. **Problem 3: Calculate the following expressions and leave answers in surd form:** **a)** $\sqrt{3}(2 + \sqrt{2}) = 2\sqrt{3} + \sqrt{3} \times \sqrt{2} = 2\sqrt{3} + \sqrt{6}$ **b)** $\sqrt{5}(3 - \sqrt{5}) = 3\sqrt{5} - \sqrt{5} \times \sqrt{5} = 3\sqrt{5} - 5$ **c)** $\sqrt{2}(4 - \sqrt{11}) = 4\sqrt{2} - \sqrt{2} \times \sqrt{11} = 4\sqrt{2} - \sqrt{22}$ **d)** $(4 + \sqrt{2})(3 - \sqrt{5}) = 4 \times 3 - 4 \times \sqrt{5} + \sqrt{2} \times 3 - \sqrt{2} \times \sqrt{5}$ $$= 12 - 4\sqrt{5} + 3\sqrt{2} - \sqrt{10}$$ **e)** $(4 + \sqrt{5})(2 + \sqrt{5}) = 4 \times 2 + 4 \times \sqrt{5} + \sqrt{5} \times 2 + \sqrt{5} \times \sqrt{5}$ $$= 8 + 4\sqrt{5} + 2\sqrt{5} + 5 = 13 + 6\sqrt{5}$$ **f)** $(4 + \sqrt{3})(2 - \sqrt{3}) = 4 \times 2 - 4 \times \sqrt{3} + \sqrt{3} \times 2 - \sqrt{3} \times \sqrt{3}$ $$= 8 - 4\sqrt{3} + 2\sqrt{3} - 3 = 5 - 2\sqrt{3}$$ **g)** $(11 + \sqrt{5})^2 = (11)^2 + 2 \times 11 \times \sqrt{5} + (\sqrt{5})^2$ $$= 121 + 22\sqrt{5} + 5 = 126 + 22\sqrt{5}$$ **h)** $(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{5}) = \sqrt{3} \times \sqrt{3} - \sqrt{3} \times \sqrt{5} + \sqrt{2} \times \sqrt{3} - \sqrt{2} \times \sqrt{5}$ $$= 3 - \sqrt{15} + \sqrt{6} - \sqrt{10}$$