ABC26GN1960 · Surds and Indices

$$\begin{aligned} & \frac{1}{\sqrt{12-\sqrt{140}}}-\frac{1}{\sqrt{8-\sqrt{60}}}-\frac{2}{\sqrt{10+\sqrt{84}}} \\ & =\frac{1}{\sqrt{12-\sqrt{4 \times 35}}}-\frac{1}{\sqrt{8-\sqrt{4 \times 15}}}-\frac{2}{\sqrt{10+\sqrt{4 \times 21}}} \\ & =\frac{1}{\sqrt{12-2 \sqrt{35}}}-\frac{1}{\sqrt{8-2 \sqrt{15}}}-\frac{2}{\sqrt{10+2 \sqrt{21}}} \\ & =\frac{1}{\sqrt{7+5-2 \sqrt{35}}}-\frac{1}{\sqrt{5+3-2 \sqrt{15}}}-\frac{2}{\sqrt{7+3+2 \sqrt{21}}} \\ & =\frac{1}{\sqrt{(\sqrt{7})^{2}+(\sqrt{5})^{2}-2 \times \sqrt{7} \times \sqrt{5}}}-\frac{1}{\sqrt{(\sqrt{5})^{2}+(\sqrt{3})^{2}-2 \times \sqrt{5} \times \sqrt{3}}} \\ & =\frac{1}{\sqrt{(\sqrt{7}-\sqrt{5})^{2}}}-\frac{1}{\sqrt{(\sqrt{5}-\sqrt{3})^{2}}}-\frac{2}{\sqrt{(\sqrt{7}+\sqrt{3})^{2}}} \\ & =\frac{1}{(\sqrt{7}-\sqrt{5})}-\frac{1}{(\sqrt{5}-\sqrt{3})}-\frac{2}{(\sqrt{7}+\sqrt{3})} \\ & =\frac{1}{\sqrt{7}-\sqrt{5}} \times \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}+\sqrt{5}}-\frac{1}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}} \\ & =\frac{\sqrt{7}+\sqrt{5}}{7-5}-\frac{\sqrt{5}+\sqrt{3}}{5-3}-\frac{2(\sqrt{7}-\sqrt{3})}{7-3} \times \frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}-\sqrt{3}} \\ & =\frac{(\sqrt{7}+\sqrt{5})}{2}-\frac{(\sqrt{5}+\sqrt{3})}{2}-\frac{(\sqrt{7}-\sqrt{3})}{2} \end{aligned}$$