Question

\(\dfrac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)

Answer 1

\(\dfrac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)

\(=\dfrac{\left(\sqrt{7}+\sqrt{5}\right)^2}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\dfrac{\left(\sqrt{7}-\sqrt{5}\right)^2}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}\)

\(=\dfrac{7+2\sqrt{35}+5+7-2\sqrt{35}+5}{7-5}\)

\(=\dfrac{14+10}{2}\)

\(=12\)

Answer 2

\(\dfrac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)

\(=\dfrac{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\dfrac{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}{\left(\sqrt{7}+\sqrt{5}\right)\left(\sqrt{7}-\sqrt{5}\right)}\)

\(=\dfrac{7+\sqrt{35}+\sqrt{35}+5}{7-5}+\dfrac{7-\sqrt{35}-\sqrt{35}+5}{7-5}\)

\(=\dfrac{7+2\sqrt{35}+5}{2}+\dfrac{7-2\sqrt{35}+5}{2}\)

\(=\dfrac{7+2\sqrt{35}+5+7-2\sqrt{35}+5}{2}\)

\(=\dfrac{24}{2}=12\)