1. **Problem Statement:** Rationalise the denominators of the following expressions and simplify where possible. 2. **Formula and Rule:** To rationalise a denominator containing a square root, multiply numerator and denominator by the same square root to eliminate the root in the denominator. 3. **Step-by-step solutions:** **a) $ \frac{1}{\sqrt{17}} $** Multiply numerator and denominator by $ \sqrt{17} $: $$ \frac{1}{\sqrt{17}} \times \frac{\sqrt{17}}{\sqrt{17}} = \frac{\sqrt{17}}{17} $$ **b) $ \frac{2}{\sqrt{2}} $** $$ \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{\cancel{\sqrt{2}}\cancel{\sqrt{2}}} = \frac{2\sqrt{2}}{2} $$ Simplify by canceling 2: $$ \frac{\cancel{2}\sqrt{2}}{\cancel{2}} = \sqrt{2} $$ **c) $ \frac{3}{\sqrt{3}} $** $$ \frac{3}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{\cancel{\sqrt{3}}\cancel{\sqrt{3}}} = \frac{3\sqrt{3}}{3} $$ Simplify by canceling 3: $$ \frac{\cancel{3}\sqrt{3}}{\cancel{3}} = \sqrt{3} $$ **d) $ \frac{10}{\sqrt{5}} $** $$ \frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{\cancel{\sqrt{5}}\cancel{\sqrt{5}}} = \frac{10\sqrt{5}}{5} $$ Simplify by dividing numerator and denominator by 5: $$ \frac{\cancel{10/5}\sqrt{5}}{\cancel{5/5}} = 2\sqrt{5} $$ **e) $ \frac{8}{\sqrt{2}} $** $$ \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{\cancel{\sqrt{2}}\cancel{\sqrt{2}}} = \frac{8\sqrt{2}}{2} $$ Simplify by dividing numerator and denominator by 2: $$ \frac{\cancel{8/2}\sqrt{2}}{\cancel{2/2}} = 4\sqrt{2} $$ **f) $ \frac{4}{\sqrt{12}} $** First simplify $ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} $, so: $$ \frac{4}{\sqrt{12}} = \frac{4}{2\sqrt{3}} = \frac{4}{2\sqrt{3}} $$ Simplify numerator and denominator by 2: $$ \frac{\cancel{4/2}}{\cancel{2/2}\sqrt{3}} = \frac{2}{\sqrt{3}} $$ Now rationalise denominator: $$ \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} $$ **Final answers:** $ \frac{\sqrt{17}}{17}, \quad \sqrt{2}, \quad \sqrt{3}, \quad 2\sqrt{5}, \quad 4\sqrt{2}, \quad \frac{2\sqrt{3}}{3} $