1. Problem a) Simplify $$(\sqrt{a} - 1 - \sqrt{a} + 1)(\sqrt{a} + 1 + \sqrt{a} - 1)$$ Step 1: Simplify inside each parenthesis: $$\sqrt{a} - 1 - \sqrt{a} + 1 = 0$$ $$\sqrt{a} + 1 + \sqrt{a} - 1 = 2\sqrt{a}$$ Step 2: Multiply the simplified terms: $$0 \times 2\sqrt{a} = 0$$ Answer a): $0$ 2. Problem b) Simplify $$2\sqrt{6} - 3\sqrt{6} - \sqrt{6} - (2\sqrt{2} - \sqrt{3})^2$$ Step 1: Combine like terms: $$2\sqrt{6} - 3\sqrt{6} - \sqrt{6} = (2 - 3 - 1)\sqrt{6} = -2\sqrt{6}$$ Step 2: Expand the square: $$(2\sqrt{2} - \sqrt{3})^2 = (2\sqrt{2})^2 - 2 \times 2\sqrt{2} \times \sqrt{3} + (\sqrt{3})^2 = 8 - 4\sqrt{6} + 3 = 11 - 4\sqrt{6}$$ Step 3: Substitute back: $$-2\sqrt{6} - (11 - 4\sqrt{6}) = -2\sqrt{6} - 11 + 4\sqrt{6} = 2\sqrt{6} - 11$$ Answer b): $2\sqrt{6} - 11$ 3. Problem c) Simplify $$(\sqrt{x^2 - 4} + \sqrt{x^2 + 4})(\sqrt{x^2 - 4} - \sqrt{x^2 + 4})$$ Step 1: Recognize difference of squares: $$(A + B)(A - B) = A^2 - B^2$$ Step 2: Apply: $$ (\sqrt{x^2 - 4})^2 - (\sqrt{x^2 + 4})^2 = (x^2 - 4) - (x^2 + 4) = -8$$ Answer c): $-8$ 4. Problem d) Simplify $$\frac{2}{3}\sqrt{5} + \sqrt{6} + \frac{1}{3}\sqrt{5} - 2\sqrt{6} - 3\sqrt{24}$$ Step 1: Combine like terms: $$\frac{2}{3}\sqrt{5} + \frac{1}{3}\sqrt{5} = \sqrt{5}$$ $$\sqrt{6} - 2\sqrt{6} = -\sqrt{6}$$ Step 2: Simplify $\sqrt{24}$: $$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ Step 3: Substitute and simplify: $$\sqrt{5} - \sqrt{6} - 3 \times 2\sqrt{6} = \sqrt{5} - \sqrt{6} - 6\sqrt{6} = \sqrt{5} - 7\sqrt{6}$$ Answer d): $\sqrt{5} - 7\sqrt{6}$ 5. Problem e) Simplify $$\sqrt{2} - \sqrt{2}(\sqrt{a} + \sqrt{b})(\sqrt{2a} - \sqrt{2b})$$ Step 1: Expand the product inside: $$(\sqrt{a} + \sqrt{b})(\sqrt{2a} - \sqrt{2b}) = \sqrt{a}\sqrt{2a} - \sqrt{a}\sqrt{2b} + \sqrt{b}\sqrt{2a} - \sqrt{b}\sqrt{2b}$$ Step 2: Simplify each term: $$\sqrt{a}\sqrt{2a} = \sqrt{2a^2} = a\sqrt{2}$$ $$\sqrt{a}\sqrt{2b} = \sqrt{2ab}$$ $$\sqrt{b}\sqrt{2a} = \sqrt{2ab}$$ $$\sqrt{b}\sqrt{2b} = b\sqrt{2}$$ Step 3: Combine terms: $$a\sqrt{2} - \sqrt{2ab} + \sqrt{2ab} - b\sqrt{2} = a\sqrt{2} - b\sqrt{2} = (a - b)\sqrt{2}$$ Step 4: Multiply by $\sqrt{2}$: $$\sqrt{2} \times (a - b)\sqrt{2} = (a - b) \times 2 = 2(a - b)$$ Step 5: Substitute back: $$\sqrt{2} - 2(a - b) = \sqrt{2} - 2a + 2b$$ Answer e): $\sqrt{2} - 2a + 2b$ 6. Problem f) Simplify $$(r + \sqrt{24})(2\sqrt{6} - r)(-1)$$ Step 1: Simplify $\sqrt{24} = 2\sqrt{6}$ Step 2: Rewrite: $$(r + 2\sqrt{6})(2\sqrt{6} - r)(-1)$$ Step 3: Multiply first two factors: $$(r)(2\sqrt{6}) - r^2 + 2\sqrt{6} \times 2\sqrt{6} - 2\sqrt{6} r = 2r\sqrt{6} - r^2 + 4 \times 6 - 2r\sqrt{6} = -r^2 + 24$$ Step 4: Multiply by $-1$: $$-(-r^2 + 24) = r^2 - 24$$ Answer f): $r^2 - 24$ 7. Problem g) Simplify $$2(x - y)(\frac{1}{\sqrt{2}} x + \sqrt{\frac{1}{2}} y)$$ Step 1: Note $\frac{1}{\sqrt{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}$ Step 2: Rewrite: $$2(x - y) \left( \frac{\sqrt{2}}{2} x + \frac{\sqrt{2}}{2} y \right) = 2(x - y) \frac{\sqrt{2}}{2} (x + y) = \sqrt{2} (x - y)(x + y)$$ Step 3: Use difference of squares: $$(x - y)(x + y) = x^2 - y^2$$ Step 4: Final expression: $$\sqrt{2} (x^2 - y^2)$$ Answer g): $\sqrt{2} (x^2 - y^2)$ 8. Problem h) Simplify $$\frac{\sqrt{49} - 16 + \sqrt{77} - \sqrt{33}}{\sqrt{11}}$$ Step 1: Simplify $\sqrt{49} = 7$ Step 2: Numerator: $$7 - 16 + \sqrt{77} - \sqrt{33} = -9 + \sqrt{77} - \sqrt{33}$$ Step 3: Write as: $$\frac{-9 + \sqrt{77} - \sqrt{33}}{\sqrt{11}} = \frac{-9}{\sqrt{11}} + \frac{\sqrt{77}}{\sqrt{11}} - \frac{\sqrt{33}}{\sqrt{11}}$$ Step 4: Simplify radicals: $$\frac{\sqrt{77}}{\sqrt{11}} = \sqrt{\frac{77}{11}} = \sqrt{7}$$ $$\frac{\sqrt{33}}{\sqrt{11}} = \sqrt{\frac{33}{11}} = \sqrt{3}$$ Step 5: Final expression: $$-\frac{9}{\sqrt{11}} + \sqrt{7} - \sqrt{3}$$ Answer h): $-\frac{9}{\sqrt{11}} + \sqrt{7} - \sqrt{3}$ 9. Problem i) Simplify $$\frac{\sqrt{k^2 - 2kg + g^2}}{|10k - 10g|}, k \neq g$$ Step 1: Recognize numerator as perfect square: $$\sqrt{(k - g)^2} = |k - g|$$ Step 2: Denominator: $$|10k - 10g| = 10|k - g|$$ Step 3: Simplify fraction: $$\frac{|k - g|}{10|k - g|} = \frac{1}{10}$$ Answer i): $\frac{1}{10}$ 10. Problem j) Simplify $$\sqrt{4x^2 + 4xy + y^2} \cdot |2x + y|^{-1}, 2x + y \neq 0$$ Step 1: Recognize perfect square inside root: $$4x^2 + 4xy + y^2 = (2x + y)^2$$ Step 2: Simplify root: $$\sqrt{(2x + y)^2} = |2x + y|$$ Step 3: Multiply by inverse: $$|2x + y| \cdot \frac{1}{|2x + y|} = 1$$ Answer j): $1$