\(A=\sqrt{5+\sqrt{21}}+\sqrt{5-\sqrt{21}}\)
\(=\sqrt{\frac{2\left(5+\sqrt{21}\right)}{2}}+\sqrt{\frac{2\left(5-\sqrt{21}\right)}{2}}\)
\(=\sqrt{\frac{10+2\sqrt{21}}{2}}+\sqrt{\frac{10-2\sqrt{21}}{2}}\)
\(=\sqrt{\frac{3+2\sqrt{21}+7}{2}}+\sqrt{\frac{3-2\sqrt{21+7}}{2}}\)
\(=\sqrt{\frac{\left(\sqrt{3}+\sqrt{7}\right)}{2}}+\sqrt{\frac{\left(\sqrt{3}+\sqrt{7}\right)}{2}}\)
\(=\frac{\left|\sqrt{3}+\sqrt{7}\right|}{2}+\frac{\left|\sqrt{3}-\sqrt{7}\right|}{2}\)
\(=\frac{\sqrt{3}+\sqrt{7}-\sqrt{3}-\sqrt{7}}{2}\)
\(=\sqrt{\frac{6}{2}}=\sqrt{3}\)
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