[tex] \sqrt{2 + \sqrt{8} } = \sqrt{2 + \sqrt{2 {}^{2} } \sqrt{2} } = \sqrt{2 + 2 \sqrt{2} } = ( \sqrt{2 + 2 \sqrt{2} } ) {}^{2} = 2 + 2 \sqrt{2} [/tex]
[tex]2 \sqrt{500} - \sqrt{3} \sqrt{1500} = 20 \sqrt{5} - \sqrt{3} \times 10 \sqrt{15} = ( \sqrt[4]{2000} - \sqrt[4]{4500} ) \times ( \sqrt[4]{2000} + \sqrt[4]{4500} ) = \sqrt[4]{125} \sqrt[4]{125} \times (2 - \sqrt{6} )(2 + \sqrt{6} ) = \sqrt[4]{15625} \times (2 - \sqrt{6} ) \times ( 2 + \sqrt{6} ) = ( \sqrt[4]{15625} \times (2 - \sqrt{6} )(2 + \sqrt{6} )) {}^{2} = (10 \sqrt{5} - 5 \sqrt[4]{25 \times 6 {}^{2} } + 2 + \sqrt{6} ) {}^{2} = (10 \sqrt{5} - 5 \sqrt[4]{30 {}^{2} } + 2 + \sqrt{6} ) {}^{2} = (10 \sqrt{5} - 5 \sqrt{30} + 2 + \sqrt{6} ) {}^{2} [/tex]
[tex]4 \sqrt{20} - 3 \sqrt{45} = 8 \sqrt{5} - 9 \sqrt{5} = - \sqrt{5} = ( - \sqrt{5}) {}^{2} = 5[/tex]
