a) \(\sqrt{26+15\sqrt{3}}\)
\(=\frac{\sqrt{52+30\sqrt{3}}}{\sqrt{2}}\)
\(=\frac{\sqrt{\left(3\sqrt{3}\right)^2+2.3\sqrt{3}.5+5^2}}{\sqrt{2}}\)
\(=\frac{\sqrt{\left(3\sqrt{3}+5\right)^2}}{\sqrt{2}}=\frac{3\sqrt{3}+5}{\sqrt{2}}\)
b) \(\)\(\sqrt{2-\sqrt{3}}=\frac{\sqrt{4-2\sqrt{3}}}{\sqrt{2}}=\frac{\sqrt{\left(\sqrt{3}-1\right)^2}}{\sqrt{2}}\)
\(=\frac{\left|\sqrt{3}-1\right|}{\sqrt{2}}=\frac{\sqrt{3}-1}{\sqrt{2}}\)
c) \(\left(\sqrt{10}-\sqrt{2}\right).\left(\sqrt{3+5}\right)\)
\(=\sqrt{10}.\sqrt{8}-\sqrt{2}.\sqrt{8}\)
\(=\sqrt{80}-\sqrt{16}=4\sqrt{5}-4\)
d) \(\left(\sqrt{6}-2\right)\left(5+\sqrt{24}\right)\sqrt{5-\sqrt{24}}\)
\(=\left(\sqrt{6}-2\right)\left(\sqrt{5+\sqrt{24}}\right).\sqrt{5-\sqrt{24}}.\left(\sqrt{5+\sqrt{24}}\right)\)
\(=\left(\sqrt{6}-2\right)\left(\sqrt{5+\sqrt{24}}\right).1\)
\(=\left(\sqrt{6}-2\right).\left(\sqrt{5+\sqrt{24}}\right)\)
\(=\sqrt{2}.\left(\sqrt{3}-\sqrt{2}\right).\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}\)
\(=\sqrt{2}.\left(3-2\right)=\sqrt{2}\)
