1. **Problem statement:** Simplify each expression by combining like surds and leaving answers in simplified surd form. 2. **Formula and rules:** - Simplify surds by factoring out perfect squares: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$ where $a$ is a perfect square. - Combine like terms: terms with the same surd can be added or subtracted. 3. **Step-by-step solutions:** a) $\sqrt{75} + \sqrt{27}$ - Simplify each surd: $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ $$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$$ - Combine like terms: $$5\sqrt{3} + 3\sqrt{3} = (5 + 3)\sqrt{3} = 8\sqrt{3}$$ b) $\sqrt{45} + \sqrt{80}$ - Simplify each surd: $$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$ $$\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$$ - Combine like terms: $$3\sqrt{5} + 4\sqrt{5} = (3 + 4)\sqrt{5} = 7\sqrt{5}$$ c) $3\sqrt{3} + \sqrt{48} - \sqrt{75}$ - Simplify each surd: $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$$ $$\sqrt{75} = 5\sqrt{3}$$ (from part a) - Substitute and combine: $$3\sqrt{3} + 4\sqrt{3} - 5\sqrt{3} = (3 + 4 - 5)\sqrt{3} = 2\sqrt{3}$$ d) $\sqrt{12} + \sqrt{27} - \sqrt{5}$ - Simplify each surd: $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ $$\sqrt{27} = 3\sqrt{3}$$ (from part a) - Combine like terms: $$2\sqrt{3} + 3\sqrt{3} - \sqrt{5} = 5\sqrt{3} - \sqrt{5}$$ (cannot combine further) e) $\sqrt{20} + \sqrt{80}$ - Simplify each surd: $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ $$\sqrt{80} = 4\sqrt{5}$$ (from part b) - Combine like terms: $$2\sqrt{5} + 4\sqrt{5} = 6\sqrt{5}$$ f) $\sqrt{28} + 3\sqrt{63} - \sqrt{7}$ - Simplify each surd: $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ $$\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$$ - Substitute and combine: $$2\sqrt{7} + 3 \times 3\sqrt{7} - \sqrt{7} = 2\sqrt{7} + 9\sqrt{7} - \sqrt{7} = (2 + 9 - 1)\sqrt{7} = 10\sqrt{7}$$ **Final answers:** a) $8\sqrt{3}$ b) $7\sqrt{5}$ c) $2\sqrt{3}$ d) $5\sqrt{3} - \sqrt{5}$ e) $6\sqrt{5}$ f) $10\sqrt{7}$