Find the value correct to three decimal places
\[ \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}}, \quad \text{where } \sqrt{2} = 1.4142 \]
Solution:
\[ \frac{1 + \sqrt{2}}{3 – 2\sqrt{2}} \times \frac{3 + 2\sqrt{2}}{3 + 2\sqrt{2}} \]
\[ = \frac{(1 + \sqrt{2})(3 + 2\sqrt{2})}{9 – 8} \]
\[ = (1 + \sqrt{2})(3 + 2\sqrt{2}) \]
\[ = 3 + 2\sqrt{2} + 3\sqrt{2} + 4 \]
\[ = 7 + 5\sqrt{2} \]
\[ = 7 + 5(1.4142) \]
\[ = 7 + 7.071 \]
\[ = 14.071 \ (\text{correct to three decimal places}) \]