Question

Помагите пожалуйста ​

Answer 1

1.103

1) $\sqrt{7.5} \: u \: \sqrt{7.6} , \: 7.5 < 7.6, \:$

$\sqrt{7.5} < \sqrt{7.6}$

2) $\sqrt{ \frac{1}{3} } \: u \: \sqrt{0.3} , \: \sqrt{ \frac{1}{3} } \: u \: \sqrt{ \frac{3}{10} } , \:$

$\sqrt{ \frac{10}{30} } \: u \: \sqrt{ \frac{9}{30} } , \: \frac{10}{30} > \frac{9}{30} ,$

$\sqrt{ \frac{1}{3} } > \sqrt{0.3}$

3) $\sqrt{0.1} \: u \: \sqrt{0.01} , \: 0.1 > 0.01,$

$\sqrt{0.1} > \sqrt{0.01}$

4) $\sqrt{2.16} \: u \: \sqrt{2 \frac{1}{6} } , \: \sqrt{ \frac{216}{100} } \: u \: \sqrt{ \frac{13}{6} },$

$\sqrt{ \frac{54}{25} } \: u \: \sqrt{ \frac{13}{6} } , \: \sqrt{ \frac{324}{150} } \: u \: \sqrt{ \frac{325}{150} } ,$

$\frac{324}{150} < \frac{325}{150} , \: \sqrt{2.16} < \sqrt{2 \frac{1}{6} }$

5) $\sqrt{7} \: u \: 2.6, \: \sqrt{7} \: u \: \sqrt{ {(2.6)}^{2} } , \:$

$\sqrt{7} \: u \: \sqrt{6.76} , \: 7 > 6.76, \sqrt{7} > 2.6$

6) $\sqrt{ \frac{5}{6} } \: u \: \sqrt{ \frac{6}{11} } , \: \sqrt{ \frac{55}{66} } \: u \: \sqrt{ \frac{36}{66} } ,$

$\frac{55}{66} > \frac{3}{66} , \sqrt{ \frac{5}{6} } > \sqrt{ \frac{6}{11} }$

7) $3.2 \: u \: \sqrt{9.8} , \: \sqrt{ {(3.2)}^{2} } \: u \: \sqrt{9.8} ,$

$\sqrt{10.24} \: u \: \sqrt{9.8} ,10.24 > 9.8,$

$3.2 > \sqrt{9.8}$

8) $\sqrt{ \frac{1}{3} } \: u \: \sqrt{0.(3)} , \sqrt{0.(3)} \: u \: \sqrt{0.(3)} ,$

$\sqrt{ \frac{1}{3} } = \sqrt{0.(3)}$

1.104

1) $\sqrt{ \frac{ {165}^{2} - {124}^{2} }{164} } = \sqrt{ \frac{(165 - 124)(165 + 124)}{164} } = \sqrt{ \frac{41 \times289 }{164} } = \sqrt{ \frac{289}{4} } = \sqrt{ \frac{ {17}^{2} }{ {2}^{2} } } = \frac{ \sqrt{ {17}^{2} } }{ \sqrt{ {2}^{2} } } = \frac{17}{2}$

2) $\sqrt{ \frac{98}{ {176}^{2} - {112}^{2} } } = \sqrt{ \frac{98}{(176 - 112)(176 + 112)} } = \sqrt{ \frac{98}{64 \times 288} } = \sqrt{ \frac{49}{32 \times 288} } = \sqrt{ \frac{49}{16 \times 576} } = \sqrt{ \frac{ {7}^{2} }{ {4}^{2} \times {24}^{2} } } = \frac{ \sqrt{ {7}^{2} } }{ \sqrt{ {4}^{2} } \times \sqrt{ {24}^{2} } } = \frac{7}{4 \times 24} = \frac{7}{96}$

3) $\sqrt{ \frac{ {149}^{2} - {76}^{2} }{ {457}^{2} - {384}^{2} } } = \sqrt{ \frac{(149 - 76)(149 + 76)}{(457 - 384)(457 + 384)} } = \sqrt{ \frac{73 \times 225}{73 \times 841} } = \sqrt{ \frac{225}{841} } =\sqrt{ \frac{{15}^{2}}{{29}^{2}} }= \frac{ \sqrt{ {15}^{2} } }{ \sqrt{ {29}^{2} } } = \frac{15}{29}$

4) $\sqrt{ \frac{ {145.5}^{2} - {96.5}^{2} }{ {193.5}^{2} - {31.5}^{2} } } = \sqrt{ \frac{(145.5 - 96.5)(145.5 + 96.5)}{(193.5 - 31.5)(193.5 + 31.5)} } = \sqrt{ \frac{49 \times 242}{162 \times 225} } = \sqrt{ \frac{49 \times 121}{81 \times 225} } = \sqrt{ \frac{ {7}^{2} \times {11}^{2} }{ {9}^{2} \times {15}^{2} } } = \frac{ \sqrt{ {7}^{2} } \times \sqrt{ {11}^{2} } }{ \sqrt{ {9}^{2} } \times \sqrt{ {15}^{2} } } = \frac{7 \times 11}{9 \times 15} = \frac{77}{135}$