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.
$frac{9}{5 - sqrt{7} } + frac{22}{7 + sqrt{5} } - frac{1}{ sqrt{7} + sqrt{5} } = frac{9(5 + sqrt{7)} }{(5 - sqrt{7})(5 + sqrt{7} )} + frac{22(7 - sqrt{5)} }{(7 + sqrt{5})(7 - sqrt{5} ) } - frac{ sqrt{7} - sqrt{5} }{( sqrt{7} + sqrt{5} )( sqrt{7} - sqrt{5}) } = frac{9( 5+ sqrt{7} )}{25 - 7} + frac{22( 7- sqrt{5}) }{49 - 5} - frac{ sqrt{7} - sqrt{5} }{7 - 5} = frac{9(5 + sqrt{7}) }{18} + frac{22(7 - sqrt{5} )}{44} - frac{ sqrt{7} - sqrt{5} }{2} = frac{5 + sqrt{7} }{2} + frac{7 - sqrt{5} }{2} - frac{ sqrt{7} - sqrt{5} }{2} = frac{5 + sqrt{7} + 7 - sqrt{5} - sqrt{7} + sqrt{5} }{2} = frac{12}{2} = 6$