1. Problem 17 (a): Simplify $\sqrt{98}$. We factor 98 as $98 = 49 \times 2$. Since $\sqrt{49} = 7$, we have: $$\sqrt{98} = \sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}.$$ 2. Problem 17 (b): Rationalize the denominator of $\frac{3}{\sqrt{5} - 2}$. Multiply numerator and denominator by the conjugate $\sqrt{5} + 2$: $$\frac{3}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{3(\sqrt{5} + 2)}{(\sqrt{5} - 2)(\sqrt{5} + 2)}.$$ The denominator becomes: $$ (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1.$$ So the expression simplifies to: $$3(\sqrt{5} + 2) = 3\sqrt{5} + 6.$$ 3. Problem 18: Simplify $\sqrt{300} + \sqrt{48}$. Factor inside the roots: $$\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3},$$ $$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}.$$ Adding these: $$10\sqrt{3} + 4\sqrt{3} = (10 + 4)\sqrt{3} = 14\sqrt{3}.$$ 4. Problem 19 (a): Simplify $22\sqrt{35} + 11\sqrt{5}$. We check if terms can be combined. Since $\sqrt{35} = \sqrt{7 \times 5} = \sqrt{7} \times \sqrt{5}$ and $\sqrt{5}$ are different radicals, they cannot be combined directly. However, we can factor out $11\sqrt{5}$: $$22\sqrt{35} + 11\sqrt{5} = 11\sqrt{5}(2\sqrt{7} + 1).$$ 5. Problem 19 (b): Simplify $\sqrt{2} \times \sqrt{10}$. Multiply under one root: $$\sqrt{2} \times \sqrt{10} = \sqrt{2 \times 10} = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}.$$ 6. Problem 20: Expand and simplify $2\sqrt{3}(2\sqrt{30} - 4\sqrt{3})$. Distribute $2\sqrt{3}$: $$2\sqrt{3} \times 2\sqrt{30} = 4\sqrt{3 \times 30} = 4\sqrt{90},$$ $$2\sqrt{3} \times (-4\sqrt{3}) = -8\sqrt{3 \times 3} = -8\sqrt{9} = -8 \times 3 = -24.$$ Simplify $\sqrt{90}$: $$\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}.$$ So $4\sqrt{90} = 4 \times 3\sqrt{10} = 12 \sqrt{10}.$ Combine terms: $$12\sqrt{10} - 24.$$