Question

Упростить иррациональные выражения

Answer 1

19.
${(2 \sqrt{12} - \sqrt{3} )}^{2} = {(2 \sqrt{12)} }^{2} - 2 \times 2 \sqrt{12} \sqrt{ 3} + ( \sqrt{3} )^{2} = 4 \times 12 - 2 \sqrt{36} + 3 = 48 - 12 + 3 = 39$
20.
$( \frac{2}{3} \sqrt{18} + \frac{3}{4} \sqrt{2} )^{2} = {( \frac{2}{3} \sqrt{18} )}^{2} + 2 \times \frac{2}{3} \sqrt{18} \times \frac{3}{4} \sqrt{2} + ( \frac{3}{4} \sqrt{2} )^{2} = \frac{4}{9} \times 18 + \sqrt{36} + \frac{9}{16} \times 2 = 4 \times 2 + 6 + \frac{9}{8} = 14 + \frac{9}{8} = \frac{112}{8} + \frac{9}{8} = \frac{121}{8}$
21.
${(1 - \sqrt{3)} }^{2} + {(1 + \sqrt{3)} }^{2} = 1 - 2 \sqrt{3} + 3 + 1 + 2 \sqrt{3} + 3 = 8$
22.
$( \sqrt{3 - \sqrt{5} } - \sqrt{3 + \sqrt{5} } )^{2} = {( \sqrt{3 - \sqrt{5} }) }^{2} - 2 \times \sqrt{3 - \sqrt{5} } \times \sqrt{3 + \sqrt{5} } + {( \sqrt{3 + \sqrt{5} }) }^{2} = 3 - \sqrt{5} - 2 \times \sqrt{9 - 5} + 3 + \sqrt{5} = 6 - 2 \times \sqrt{4} = 6 - 4 = 2$
23.
$\frac{7}{7 + 4 \sqrt{3} } + \frac{1}{7 - 4 \sqrt{3} } = \frac{7 - 4 \sqrt{3} + 7 + 4 \sqrt{3} }{(7 + 4 \sqrt{3} )(7 - 4 \sqrt{3} )} = \frac{14}{49 - 16 \times 3} = \frac{14}{49 - 48} = 14$
24.
$\frac{0.625 \times 6.75 {}^{2} - 3.25 {}^{2} \times 0.625 }{ \sqrt{3.5 {}^{2} + 7 \times 2.75 + 2.75 {}^{2} } } = \frac{0.625(6.75 {}^{2} - 3.25 {}^{2}) }{ \sqrt{(3.5 + 2.75) {}^{2} } } = \frac{0.625(6.75 - 3.25)(6.75 + 3.25)}{3.5 + 2.75} = \frac{0.625 \times 3.5 \times 10}{6.25} = \frac{6.25 \times 3.5}{6.25} = 3.5$
25.
$\frac{9}{5 - \sqrt{7} } + \frac{22}{7 + \sqrt{5} } - \frac{1}{ \sqrt{7} + \sqrt{5} } = \frac{9(5 + \sqrt{7)} }{(5 - \sqrt{7})(5 + \sqrt{7} )} + \frac{22(7 - \sqrt{5)} }{(7 + \sqrt{5})(7 - \sqrt{5} ) } - \frac{ \sqrt{7} - \sqrt{5} }{( \sqrt{7} + \sqrt{5} )( \sqrt{7} - \sqrt{5}) } = \frac{9( 5+ \sqrt{7} )}{25 - 7} + \frac{22( 7- \sqrt{5}) }{49 - 5} - \frac{ \sqrt{7} - \sqrt{5} }{7 - 5} = \frac{9(5 + \sqrt{7}) }{18} + \frac{22(7 - \sqrt{5} )}{44} - \frac{ \sqrt{7} - \sqrt{5} }{2} = \frac{5 + \sqrt{7} }{2} + \frac{7 - \sqrt{5} }{2} - \frac{ \sqrt{7} - \sqrt{5} }{2} = \frac{5 + \sqrt{7} + 7 - \sqrt{5} - \sqrt{7} + \sqrt{5} }{2} = \frac{12}{2} = 6$