\(M=\dfrac{4+\sqrt{7}}{\sqrt{14}+\sqrt{4+\sqrt{7}}}-\dfrac{4-\sqrt{7}}{\sqrt{14}+\sqrt{4-\sqrt{7}}}\)
\(=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)}{\sqrt{28}+\sqrt{8+2\sqrt{7}}}-\dfrac{4\sqrt{2}-\sqrt{14}}{\sqrt{28}+\sqrt{8-2\sqrt{7}}}\)
\(=\dfrac{4\sqrt{2}+\sqrt{14}}{2\sqrt{7}+\sqrt{\left(\sqrt{7}+1\right)^2}}-\dfrac{4\sqrt{2}-\sqrt{14}}{2\sqrt{7}+\sqrt{\left(\sqrt{7}-1\right)^2}}\)
\(=\dfrac{4\sqrt{2}+\sqrt{14}}{2\sqrt{7}+\sqrt{7}+1}-\dfrac{4\sqrt{2}-\sqrt{14}}{2\sqrt{7}+\sqrt{7}-1}\)
\(=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)}{3\sqrt{7}+1}-\dfrac{\sqrt{2}\left(4-\sqrt{7}\right)}{3\sqrt{7}-1}\)
\(=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)\left(3\sqrt{7}-1\right)-\sqrt{2}\left(4-\sqrt{7}\right)\left(3\sqrt{7}+1\right)}{\left(3\sqrt{7}-1\right)\left(3\sqrt{7}+1\right)}\)
\(=\dfrac{\sqrt{2}\left(12\sqrt{7}-4+21-\sqrt{7}\right)-\sqrt{2}\left(12\sqrt{7}+4-21-\sqrt{7}\right)}{63-1}\)
\(=\dfrac{\sqrt{2}\left(11\sqrt{7}+17-11\sqrt{7}+17\right)}{62}=\dfrac{34\sqrt{2}}{62}\)
\(=\dfrac{17}{31}\sqrt{2}\)
=>b=31; a=17
=>b-a=14
