Question

6-4 root 3 / 6+4 root 3

Answer 1

$$\begin{gathered} \frac{6-4\sqrt{3}}{6+4\sqrt{3}} \\ =\frac{6-4\sqrt{3}}{6+4\sqrt{3}}\times \frac{6-4\sqrt{3}}{6-4\sqrt{3}}\left[\mathrm{on}\mathrm{rationalising}\right] \\ =\frac{\left(6-4\sqrt{3}\right)\left(6-4\sqrt{3}\right)}{\left(6+4\sqrt{3}\right)\left(6-4\sqrt{3}\right)} \\ =\frac{{\left(6-4\sqrt{3}\right)}^2}{{\left(6\right)}^2-{\left(4\sqrt{3}\right)}^2} \\ =\frac{{\left(6\right)}^2+{\left(4\sqrt{3}\right)}^2-2\left(6\right)\left(4\sqrt{3}\right)}{36-48} \\ =\frac{36+48-48\sqrt{3}}{-12} \\ =\frac{84-48\sqrt{3}}{-12} \\ =\frac{48\sqrt{3}-84}{12} \\ =\frac{12\left(4\sqrt{3}-7\right)}{12} \\ =4\sqrt{3}-7 \end{gathered}$$