Question

\(\sqrt{12-2\sqrt{35}}\) - \(\sqrt{12+2\sqrt{35}}\)

Answer 1

\(\sqrt{12-2\sqrt{35}}-\sqrt{12+2\sqrt{35}}\\ =\sqrt{\left(\sqrt{5}\right)^2-2\cdot\sqrt{5}\cdot\sqrt{7}+\left(\sqrt{7}\right)^2}-\sqrt{\left(\sqrt{5}\right)^2+2\cdot\sqrt{5}\cdot\sqrt{7}+\left(\sqrt{7}\right)^2}\\ =\sqrt{\left(\sqrt{5}-\sqrt{7}\right)^2}-\sqrt{\left(\sqrt{5}+\sqrt{7}\right)^2}\\ =\left|\sqrt{5}-\sqrt{7}\right|-\left|\sqrt{5}+\sqrt{7}\right|\\ =\left(\sqrt{7}-\sqrt{5}\right)-\left(\sqrt{5}+\sqrt{7}\right)\\ =\sqrt{7}-\sqrt{5}-\sqrt{5}-\sqrt{7}\\ =-2\sqrt{5}\)