\(M=\dfrac{4+\sqrt{7}}{\sqrt{14}+\sqrt{4+\sqrt{7}}}-\dfrac{4-\sqrt{7}}{\sqrt{14}+\sqrt{4-\sqrt{7}}}\)
=>\(M=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)}{\sqrt{28}+\sqrt{8+2\sqrt{7}}}-\dfrac{\sqrt{2}\left(4-\sqrt{7}\right)}{\sqrt{28}+\sqrt{8-2\sqrt{7}}}\)
\(=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)}{2\sqrt{7}+\sqrt{\left(\sqrt{7}+1\right)^2}}-\dfrac{\sqrt{2}\left(4-\sqrt{7}\right)}{2\sqrt{7}+\sqrt{\left(\sqrt{7}-1\right)^2}}\)
\(=\dfrac{\sqrt{2}\left(4+\sqrt{7}\right)}{3\sqrt{7}+1}-\dfrac{\sqrt{2}\left(4-\sqrt{7}\right)}{3\sqrt{7}-1}\)
\(=\dfrac{\sqrt{2}\left[\left(4+\sqrt{7}\right)\left(3\sqrt{7}-1\right)-\left(4-\sqrt{7}\right)\left(3\sqrt{7}+1\right)\right]}{\left(3\sqrt{7}+1\right)\left(3\sqrt{7}-1\right)}\)
\(=\dfrac{\sqrt{2}\left[12\sqrt{7}-4+21-\sqrt{7}-12\sqrt{7}-4+21+\sqrt{7}\right]}{49-1}\)
\(=\dfrac{\sqrt{2}\left(42-8\right)}{48}=\dfrac{34\sqrt{2}}{48}=\dfrac{17\sqrt{2}}{24}\)
=>a=17; b=24
=>b-a=7
