Simplify: \(\dfrac{1}{\sqrt{7}-\sqrt{3}} + \dfrac{1}{\sqrt{7}+\sqrt{3}}\)
\(\dfrac{\sqrt{7}}{2}\)
Combine the two fractions over a common denominator:
\(\dfrac{1}{\sqrt{7}-\sqrt{3}} + \dfrac{1}{\sqrt{7}+\sqrt{3}} = \dfrac{(\sqrt{7}+\sqrt{3}) + (\sqrt{7}-\sqrt{3})}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}\)
The numerator simplifies and the denominator is a difference of squares:
\(= \dfrac{2\sqrt{7}}{(\sqrt{7})^2 - (\sqrt{3})^2} = \dfrac{2\sqrt{7}}{7 - 3} = \dfrac{2\sqrt{7}}{4} = \dfrac{\sqrt{7}}{2}\)
Hence the value is \(\dfrac{\sqrt{7}}{2}\) — option (3).
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