Mathematic Problems
$\left(\sqrt[3]{1+2\sqrt{6}}-\sqrt[6]{25+4\sqrt{6}}\right)·\sqrt[3]{2\sqrt{6}-1}$
$=\sqrt[3]{1+2\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{{1}{+}{2}\sqrt{6}}·\sqrt[3]{{2}\sqrt{6}{-}{1}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{{2}\sqrt{6}{+}{1}}·\sqrt[3]{{2}\sqrt{6}{-}{1}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{{\left(2\sqrt{6}\right)}^{{2}}{-}{{1}}^{{2}}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{{4}{·}{6}{-}{{1}}^{{}^{2}}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{{23}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$

$=\sqrt[3]{23}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[{2}{·}{3}]{{\left(2\sqrt{6}-1\right)}^{{2}}}$

$=\sqrt[3]{24-1}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[6]{{\left(2\sqrt{6}\right)}^{{2}}{-}{2}{·}{2}{·}\sqrt{6}{·}{1}{+}{1}}$

$=\sqrt[3]{24-1}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[6]{{24}{-}{4}\sqrt{6}{+}{1}}$

$=\sqrt[3]{23}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[6]{{25}{-}{4}\sqrt{6}}$

$=\sqrt[3]{23}-\sqrt[6]{{25}{+}{4}\sqrt{6}{·}}\sqrt[6]{{25}{-}{4}\sqrt{6}}$

$=\sqrt[3]{23}-\sqrt[6]{{{25}}^{{2}}{-}{\left(4\sqrt{6}\right)}^{{2}}}$

$=\sqrt[3]{23}-\sqrt[6]{{625}{-}{96}}$

$=\sqrt[3]{23}-\sqrt[6]{{529}}$

$=\sqrt[3]{23}-\sqrt[6]{{{23}}^{2}}$

$=\sqrt[3]{23}-\sqrt[{2}{·}{3}]{{23}^{{2}}}$

$=\sqrt[3]{23}-\sqrt[3]{23}$
$=0$