# Exercise ID91

Algebra → Roots → Simplifying
[Level: ] [Number of helps: 1] [Number of pictures: 0] [Number of steps: 11] [Number of characters: 0]

Evaluate the following expression:

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

$\left(\sqrt[3]{1+2\sqrt{6}}-\sqrt[6]{25+4\sqrt{6}}\right)·\sqrt[3]{2\sqrt{6}-1}$ $=\sqrt[3]{1+2\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$ $=\sqrt[3]{{1}{+}{2}\sqrt{6}}·\sqrt[3]{{2}\sqrt{6}{-}{1}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$ $=\sqrt[3]{{\left(2\sqrt{6}\right)}^{2}{-}{{1}}^{{}^{2}}}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$ $=\sqrt[3]{23}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[3]{2\sqrt{6}-1}$ $=\sqrt[3]{23}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[{6}]{{\left(2\sqrt{6}-1\right)}^{{2}}}$ $=\sqrt[3]{24-1}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[6]{24-4\sqrt{6}+1}$ $=\sqrt[3]{23}-\sqrt[6]{25+4\sqrt{6}}·\sqrt[6]{25-4\sqrt{6}}$ $=\sqrt[3]{23}-\sqrt[6]{{25}{+}{4}\sqrt{6}{·}}\sqrt[6]{{25}{-}{4}\sqrt{6}}$ $=\sqrt[3]{23}-\sqrt[6]{{25}^{2}-{\left(4\sqrt{6}\right)}^{2}}$ $=\sqrt[3]{23}-\sqrt[6]{625-96}$ $=\sqrt[3]{23}-\sqrt[6]{529}$ $=\sqrt[3]{23}-\sqrt[6]{{{23}}^{2}}$ $=\sqrt[3]{23}-\sqrt[3]{23}$ $=0$
HELP AVAILABLE!
$\left(\sqrt[3]{1+2\sqrt{6}}-\sqrt[6]{25+4\sqrt{6}}\right)·\sqrt[3]{2\sqrt{6}-1}=0$

$\sqrt[n]{a·b}=\sqrt[n]{a}·\sqrt[n]{b}$

$\sqrt[n]{{a}^{m}}=\sqrt[c·n]{{a}^{c·m}}$