EXERCISE 1.7 R \( \begin{array}{llll}\text { (a) Rationalise the denominator and leave the answer in simplest surd form: } \\ \begin{array}{llll}\text { (1) } \frac{3}{\sqrt{6}} & \text { (2) } \frac{2}{5 \sqrt{2}} & \text { (3) } \frac{3}{\sqrt[3]{9}} & \text { (4) } \frac{2}{4+\sqrt{2}}\end{array}\end{array}> \)

Certainly! Let's rationalize the denominators of each expression and present the answers in their simplest surd forms. ### (a) Rationalize the denominator and simplify: #### (1) \( \frac{3}{\sqrt{6}} \) **Solution:** To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{6} \): \[ \frac{3}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{6}}{6} = \frac{\sqrt{6}}{2} \] **Answer:** \[ \frac{\sqrt{6}}{2} \] --- #### (2) \( \frac{2}{5\sqrt{2}} \) **Solution:** Multiply both the numerator and the denominator by \( \sqrt{2} \): \[ \frac{2}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{5 \times 2} = \frac{\sqrt{2}}{5} \] **Answer:** \[ \frac{\sqrt{2}}{5} \] --- #### (3) \( \frac{3}{\sqrt[3]{9}} \) **Solution:** First, recognize that \( \sqrt[3]{9} = 9^{1/3} \). To rationalize the denominator, multiply by \( 9^{2/3} \) (which is \( \sqrt[3]{81} \)) to make the denominator a whole number: \[ \frac{3}{9^{1/3}} \times \frac{9^{2/3}}{9^{2/3}} = \frac{3 \times 9^{2/3}}{9^{1/3} \times 9^{2/3}} = \frac{3 \times 9^{2/3}}{9} = \frac{9^{2/3}}{3} \] Since \( 9^{2/3} = (3^2)^{2/3} = 3^{4/3} \), we have: \[ \frac{3^{4/3}}{3} = 3^{4/3 - 1} = 3^{1/3} = \sqrt[3]{3} \] **Answer:** \[ \sqrt[3]{3} \] --- #### (4) \( \frac{2}{4 + \sqrt{2}} \) **Solution:** To rationalize the denominator, multiply by the conjugate of the denominator, which is \( 4 - \sqrt{2} \): \[ \frac{2}{4 + \sqrt{2}} \times \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{2(4 - \sqrt{2})}{(4)^2 - (\sqrt{2})^2} = \frac{8 - 2\sqrt{2}}{16 - 2} = \frac{8 - 2\sqrt{2}}{14} \] Simplify by dividing numerator and denominator by 2: \[ \frac{8 - 2\sqrt{2}}{14} = \frac{4 - \sqrt{2}}{7} \] **Answer:** \[ \frac{4 - \sqrt{2}}{7} \] --- ### Summary of Answers: 1. \( \frac{3}{\sqrt{6}} = \frac{\sqrt{6}}{2} \) 2. \( \frac{2}{5\sqrt{2}} = \frac{\sqrt{2}}{5} \) 3. \( \frac{3}{\sqrt[3]{9}} = \sqrt[3]{3} \) 4. \( \frac{2}{4 + \sqrt{2}} = \frac{4 - \sqrt{2}}{7} \) If you have any more questions or need further clarification, feel free to ask!