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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 {(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](https://tex.z-dn.net/?f=+%7B%282+sqrt%7B12%7D++-++sqrt%7B3%7D+%29%7D%5E%7B2%7D++%3D++%7B%282+sqrt%7B12%29%7D+%7D%5E%7B2%7D++-+2+times+2+sqrt%7B12%7D++sqrt%7B+3%7D++%2B+%28+sqrt%7B3%7D+%29%5E%7B2%7D++%3D+4+times+12++-+2+sqrt%7B36%7D++%2B+3+%3D+48+-+12+%2B+3+%3D+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} ( 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}](https://tex.z-dn.net/?f=%28+frac%7B2%7D%7B3%7D++sqrt%7B18%7D++%2B++frac%7B3%7D%7B4%7D++sqrt%7B2%7D+%29%5E%7B2%7D++%3D++%7B%28+frac%7B2%7D%7B3%7D+sqrt%7B18%7D++%29%7D%5E%7B2%7D++%2B+2+times++frac%7B2%7D%7B3%7D++sqrt%7B18%7D+++times+frac%7B3%7D%7B4%7D++sqrt%7B2%7D++%2B+%28+frac%7B3%7D%7B4%7D++sqrt%7B2%7D+%29%5E%7B2%7D++%3D++frac%7B4%7D%7B9%7D++times+18+%2B+++sqrt%7B36%7D++%2B++frac%7B9%7D%7B16%7D++times+2+%3D+4+times+2+%2B+6+%2B++frac%7B9%7D%7B8%7D++%3D+14+%2B++frac%7B9%7D%7B8%7D++%3D++frac%7B112%7D%7B8%7D++%2B++frac%7B9%7D%7B8%7D++%3D++frac%7B121%7D%7B8%7D+)
21.
![{(1 - sqrt{3)} }^{2} + {(1 + sqrt{3)} }^{2} = 1 - 2 sqrt{3} + 3 + 1 + 2 sqrt{3} + 3 = 8 {(1 - sqrt{3)} }^{2} + {(1 + sqrt{3)} }^{2} = 1 - 2 sqrt{3} + 3 + 1 + 2 sqrt{3} + 3 = 8](https://tex.z-dn.net/?f=+%7B%281+-++sqrt%7B3%29%7D+%7D%5E%7B2%7D++%2B++%7B%281+%2B++sqrt%7B3%29%7D+%7D%5E%7B2%7D++%3D+1+-+2+sqrt%7B3%7D+%2B+3+%2B+1+%2B+2+sqrt%7B3%7D+++%2B+3+%3D+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 ( 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](https://tex.z-dn.net/?f=%28+sqrt%7B3+-++sqrt%7B5%7D+%7D++-++sqrt%7B3+%2B++sqrt%7B5%7D+%7D+%29%5E%7B2%7D++%3D++%7B%28+sqrt%7B3+-++sqrt%7B5%7D+%7D%29+%7D%5E%7B2%7D++-+2+times++sqrt%7B3+-++sqrt%7B5%7D+%7D++times++sqrt%7B3++%2B+++sqrt%7B5%7D+%7D++%2B++%7B%28+sqrt%7B3+%2B++sqrt%7B5%7D+%7D%29+%7D%5E%7B2%7D++%3D+3+-++sqrt%7B5%7D++-+2+times++sqrt%7B9+-+5%7D++%2B+3+%2B++sqrt%7B5%7D++%3D+6+-+2+times++sqrt%7B4%7D++%3D+6+-+4+%3D+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 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](https://tex.z-dn.net/?f=+frac%7B7%7D%7B7+%2B+4+sqrt%7B3%7D+%7D++%2B++frac%7B1%7D%7B7+-+4+sqrt%7B3%7D+%7D++%3D++frac%7B7+-+4+sqrt%7B3%7D++%2B+7+%2B+4+sqrt%7B3%7D+%7D%7B%287+%2B+4+sqrt%7B3%7D+%29%287+-+4+sqrt%7B3%7D+%29%7D++%3D+++frac%7B14%7D%7B49+-+16+times+3%7D++%3D++frac%7B14%7D%7B49+-+48%7D++%3D+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 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](https://tex.z-dn.net/?f=+frac%7B0.625+times+6.75+%7B%7D%5E%7B2%7D+-+3.25+%7B%7D%5E%7B2%7D++times+0.625+%7D%7B+sqrt%7B3.5+%7B%7D%5E%7B2%7D++%2B+7+times+2.75+%2B+2.75+%7B%7D%5E%7B2%7D+%7D+%7D++%3D++frac%7B0.625%286.75+%7B%7D%5E%7B2%7D++-+3.25+%7B%7D%5E%7B2%7D%29+%7D%7B+sqrt%7B%283.5+%2B+2.75%29+%7B%7D%5E%7B2%7D+%7D+%7D++%3D++frac%7B0.625%286.75+-+3.25%29%286.75+%2B+3.25%29%7D%7B3.5+%2B+2.75%7D++%3D++frac%7B0.625+times+3.5+times+10%7D%7B6.25%7D++%3D+++frac%7B6.25+times+3.5%7D%7B6.25%7D++%3D+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 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](https://tex.z-dn.net/?f=+frac%7B9%7D%7B5+-++sqrt%7B7%7D+%7D++%2B++frac%7B22%7D%7B7+%2B++sqrt%7B5%7D+%7D++-++frac%7B1%7D%7B+sqrt%7B7%7D++%2B++sqrt%7B5%7D+%7D+%3D++frac%7B9%285+%2B++sqrt%7B7%29%7D+%7D%7B%285+-++sqrt%7B7%7D%29%285+%2B++sqrt%7B7%7D++%29%7D++%2B++frac%7B22%287+-++sqrt%7B5%29%7D+%7D%7B%287+%2B++sqrt%7B5%7D%29%287+-++sqrt%7B5%7D+%29+%7D+++-++frac%7B+sqrt%7B7%7D++-++sqrt%7B5%7D+%7D%7B%28+sqrt%7B7%7D++%2B++sqrt%7B5%7D+%29%28+sqrt%7B7%7D++-++sqrt%7B5%7D%29+%7D++%3D++frac%7B9%28+5%2B+sqrt%7B7%7D+%29%7D%7B25+-+7%7D++%2B++frac%7B22%28+7-++sqrt%7B5%7D%29+%7D%7B49+-+5%7D++-++frac%7B+sqrt%7B7%7D+-++sqrt%7B5%7D++%7D%7B7+-+5%7D++%3D++frac%7B9%285+%2B++sqrt%7B7%7D%29+%7D%7B18%7D++%2B++frac%7B22%287+-++sqrt%7B5%7D+%29%7D%7B44%7D++-++frac%7B+sqrt%7B7%7D+-++sqrt%7B5%7D++%7D%7B2%7D++%3D++frac%7B5+%2B++sqrt%7B7%7D+%7D%7B2%7D++%2B++frac%7B7+-++sqrt%7B5%7D+%7D%7B2%7D++-++frac%7B+sqrt%7B7%7D++-++sqrt%7B5%7D+%7D%7B2%7D++%3D++frac%7B5+%2B++sqrt%7B7%7D++%2B+7+-++sqrt%7B5%7D+-++sqrt%7B7%7D++%2B++sqrt%7B5%7D++%7D%7B2%7D++%3D++frac%7B12%7D%7B2%7D++%3D+6)
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