[tex]( \sqrt{18} + \sqrt{27} ) \div 3 = (3 \sqrt{2} + 3 \sqrt{3} ) \div 3 = \frac{3 \sqrt{2} + 3 \sqrt{3} }{3} = \frac{3( \sqrt{2} + \sqrt{3}) }{3} = \sqrt{2} + \sqrt{3} [/tex]
[tex]( \sqrt{8} - \sqrt{10} ) \div \sqrt{2} = \frac{2 \sqrt{2} - \sqrt{10} }{ \sqrt{2} } = \frac{(2 \sqrt{2} - \sqrt{10}) \sqrt{2} }{ \sqrt{2} \sqrt{2} } = \frac{4 - \sqrt{20} }{2} = \frac{4 - 2 \sqrt{5} }{2} = \frac{2(2 - \sqrt{5}) }{2} = 2 - \sqrt{5} [/tex]
[tex]( \sqrt{3} - \sqrt{6} + \sqrt{15} ) \div ( - \sqrt{3} ) = - \frac{( \sqrt{3} - \sqrt{6} + \sqrt{15}) \sqrt{3} }{3} = - \frac{3 - \sqrt{18} + \sqrt{45} }{3} = - \frac{3(1 - \sqrt{2} + \sqrt{5} )}{3} = - (1 - \sqrt{2} + \sqrt{5} ) = - 1 + \sqrt{2} - \sqrt{5} [/tex]
[tex](4 \sqrt{10} + 2 \sqrt{15} ) \div (2 \sqrt{5} ) = \frac{2(2 \sqrt{10} + \sqrt{15} )}{2 \sqrt{5} } = \frac{(2 \sqrt{10} + \sqrt{15}) \sqrt{5} }{5} = \frac{10 \sqrt{2} + 5 \sqrt{3} }{5} = \frac{5(2 \sqrt{2} + \sqrt{3} )}{5} = 2 \sqrt{2} + \sqrt{3} [/tex]
[tex]( \sqrt{2} + \sqrt{6} + \sqrt{14} ) \div \frac{1}{ \sqrt{2} } = ( \sqrt{2} + \sqrt{6} + \sqrt{14} ) \sqrt{2} = 2 + \sqrt{12} + \sqrt{28} = 2 + 2 \sqrt{3} + 2 \sqrt{7} [/tex]
[tex]( \sqrt{18} - \sqrt{45} ) \div ( - \frac{3}{ \sqrt{2} } ) = (3 \sqrt{2} - 3 \sqrt{5} ) \times ( - \frac{ \sqrt{2} }{3}) = - \frac{6 - 3 \sqrt{10} }{3} = - (2 - \sqrt{10} ) = - 2 + \sqrt{10} [/tex]
