Find the value correct to three decimal places
\[ \frac{4}{3\sqrt{3} – 2\sqrt{2}} + \frac{3}{3\sqrt{3} + 2\sqrt{2}}, \quad \text{where } \sqrt{2} = 1.4142,\ \sqrt{3} = 1.732 \]
Solution:
\[ \frac{4(3\sqrt{3} + 2\sqrt{2}) + 3(3\sqrt{3} – 2\sqrt{2})}{(3\sqrt{3})^2 – (2\sqrt{2})^2} \]
\[ = \frac{12\sqrt{3} + 8\sqrt{2} + 9\sqrt{3} – 6\sqrt{2}}{27 – 8} \]
\[ = \frac{21\sqrt{3} + 2\sqrt{2}}{19} \]
\[ = \frac{21(1.732) + 2(1.4142)}{19} \]
\[ = \frac{36.372 + 2.8284}{19} \]
\[ = \frac{39.2004}{19} \]
\[ = 2.063 \ (\text{correct to three decimal places}) \]