i) \(\sqrt{8-3\sqrt{7}}+\sqrt{4-\sqrt{7}}=\sqrt{\dfrac{16-6\sqrt{7}}{2}}+\sqrt{\dfrac{8-2\sqrt{7}}{2}}\)
\(=\sqrt{\dfrac{\left(3-\sqrt{7}\right)^2}{2}}+\sqrt{\dfrac{\left(\sqrt{7}-1\right)^2}{2}}=\dfrac{\left|3-\sqrt{7}\right|}{\sqrt{2}}+\dfrac{\left|\sqrt{7}-1\right|}{\sqrt{2}}\)
\(=\dfrac{3-\sqrt{7}}{\sqrt{2}}+\dfrac{\sqrt{7}-1}{\sqrt{2}}=\dfrac{2}{\sqrt{2}}=\sqrt{2}\)
j) \(\sqrt{5+\sqrt{21}}-\sqrt{5-\sqrt{21}}=\sqrt{\dfrac{10+2\sqrt{21}}{2}}-\sqrt{\dfrac{10-2\sqrt{21}}{2}}\)
\(=\sqrt{\dfrac{\left(\sqrt{7}+\sqrt{3}\right)^2}{2}}-\sqrt{\dfrac{\left(\sqrt{7}-\sqrt{3}\right)^2}{2}}=\dfrac{\left|\sqrt{7}+\sqrt{3}\right|}{\sqrt{2}}-\dfrac{\left|\sqrt{7}-\sqrt{3}\right|}{\sqrt{2}}\)
\(=\dfrac{\sqrt{7}+\sqrt{3}}{\sqrt{2}}-\dfrac{\sqrt{7}-\sqrt{3}}{\sqrt{2}}=\dfrac{2\sqrt{3}}{\sqrt{2}}=\sqrt{6}\)
