1. **Problem:** Rationalise the denominators of the given fractions and simplify where possible. 2. **Formula and rule:** To rationalise a denominator with a square root, multiply numerator and denominator by the same square root to make the denominator a rational number. 3. **Step-by-step solutions:** **a) $ \frac{1}{\sqrt{2}} $** Multiply numerator and denominator by $ \sqrt{2} $: $$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{\cancel{\sqrt{2}} \times \sqrt{2}} = \frac{\sqrt{2}}{2}$$ **b) $ \frac{1}{\sqrt{19}} $** Multiply numerator and denominator by $ \sqrt{19} $: $$\frac{1}{\sqrt{19}} \times \frac{\sqrt{19}}{\sqrt{19}} = \frac{\sqrt{19}}{\cancel{\sqrt{19}} \times \sqrt{19}} = \frac{\sqrt{19}}{19}$$ **c) $ \frac{\sqrt{3}}{\sqrt{15}} $** Multiply numerator and denominator by $ \sqrt{15} $: $$\frac{\sqrt{3}}{\sqrt{15}} \times \frac{\sqrt{15}}{\sqrt{15}} = \frac{\sqrt{3} \times \sqrt{15}}{\cancel{\sqrt{15}} \times \sqrt{15}} = \frac{\sqrt{45}}{15}$$ Simplify $ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} $: $$\frac{3\sqrt{5}}{15} = \frac{\cancel{3}\sqrt{5}}{\cancel{15}} = \frac{\sqrt{5}}{5}$$ **d) $ \frac{\sqrt{6}}{\sqrt{48}} $** Multiply numerator and denominator by $ \sqrt{48} $: $$\frac{\sqrt{6}}{\sqrt{48}} \times \frac{\sqrt{48}}{\sqrt{48}} = \frac{\sqrt{6} \times \sqrt{48}}{\cancel{\sqrt{48}} \times \sqrt{48}} = \frac{\sqrt{288}}{48}$$ Simplify $ \sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2} $: $$\frac{12\sqrt{2}}{48} = \frac{\cancel{12}\sqrt{2}}{\cancel{48}} = \frac{\sqrt{2}}{4}$$ **e) $ \frac{7}{\sqrt{63}} $** Multiply numerator and denominator by $ \sqrt{63} $: $$\frac{7}{\sqrt{63}} \times \frac{\sqrt{63}}{\sqrt{63}} = \frac{7\sqrt{63}}{\cancel{\sqrt{63}} \times \sqrt{63}} = \frac{7\sqrt{63}}{63}$$ Simplify $ \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} $: $$\frac{7 \times 3\sqrt{7}}{63} = \frac{21\sqrt{7}}{63} = \frac{\cancel{21}\sqrt{7}}{\cancel{63}} = \frac{\sqrt{7}}{3}$$ **f) $ \frac{\sqrt{12}}{\sqrt{156}} $** Multiply numerator and denominator by $ \sqrt{156} $: $$\frac{\sqrt{12}}{\sqrt{156}} \times \frac{\sqrt{156}}{\sqrt{156}} = \frac{\sqrt{12} \times \sqrt{156}}{\cancel{\sqrt{156}} \times \sqrt{156}} = \frac{\sqrt{1872}}{156}$$ Simplify $ \sqrt{1872} = \sqrt{144 \times 13} = 12\sqrt{13} $: $$\frac{12\sqrt{13}}{156} = \frac{\cancel{12}\sqrt{13}}{\cancel{156}} = \frac{\sqrt{13}}{13}$$ 4. **Final answers:** - a) $ \frac{\sqrt{2}}{2} $ - b) $ \frac{\sqrt{19}}{19} $ - c) $ \frac{\sqrt{5}}{5} $ - d) $ \frac{\sqrt{2}}{4} $ - e) $ \frac{\sqrt{7}}{3} $ - f) $ \frac{\sqrt{13}}{13} $