ABC26GN1961 · Surds and Indices

$\sqrt{4+\sqrt{15}}=\sqrt{\frac{5}{2}+\frac{3}{2}+2 \times \frac{\sqrt{5}}{\sqrt{2}} \times \frac{\sqrt{3}}{\sqrt{2}}}=\sqrt{\left(\frac{\sqrt{5}}{\sqrt{2}}\right)^{2}+\left(\frac{\sqrt{3}}{\sqrt{2}}\right)^{2}+2 \times \frac{\sqrt{5}}{\sqrt{2}} \times \frac{\sqrt{3}}{\sqrt{2}}}=\sqrt{\left(\frac{\sqrt{5}}{\sqrt{2}}+\frac{\sqrt{3}}{\sqrt{2}}\right)^{2}}=\frac{\sqrt{5}}{\sqrt{2}}+\frac{\sqrt{3}}{\sqrt{2}}. \text{ Similarly, } \sqrt{4-\sqrt{15}}=\frac{\sqrt{5}}{\sqrt{2}}-\frac{\sqrt{3}}{\sqrt{2}}. \sqrt{12-4 \sqrt{5}}=\sqrt{10+2-2 \times \sqrt{10} \times \sqrt{2}}=\sqrt{(\sqrt{10})^{2}+(\sqrt{2})^{2}-2 \times \sqrt{10} \times \sqrt{2}}=\sqrt{(\sqrt{10}-\sqrt{2})^{2}}=(\sqrt{10}-\sqrt{2}). \therefore \text{ Given expression } =\left(\frac{\sqrt{5}}{\sqrt{2}}+\frac{\sqrt{3}}{\sqrt{2}}\right)+\left(\frac{\sqrt{5}}{\sqrt{2}}-\frac{\sqrt{3}}{\sqrt{2}}\right)-(\sqrt{10}-\sqrt{2})=\frac{2 \sqrt{5}}{\sqrt{2}}-\sqrt{10}+\sqrt{2}=\sqrt{10}-\sqrt{10}+\sqrt{2}=\sqrt{2},$ which is an irrational number.