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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$=\sqrt[3]{23}-\sqrt[2·3]{{23}^2}$$