Question

media armonica pentru nr :
x=
$( \sqrt{7} - 5 \sqrt{3} ) ^{2}$
y=
$( \sqrt{7} + 5 \sqrt{3} ) ^{2}$

Answer 1

$x = {(\sqrt{7} - 5 \sqrt{3} )}^{2}$

$x = ({\sqrt{7})}^{2} - 2 \times \sqrt{7} \times 5 \sqrt{3} + {(5 \sqrt{3}) }^{2}$

$x = 7 - 10 \sqrt{21} + 75$

$x = 82 - 10 \sqrt{21}$

$y = {(\sqrt{7} + 5 \sqrt{3}) }^{2}$

$y = ({\sqrt{7})}^{2} + 2 \times \sqrt{7} \times 5 \sqrt{3} + {(5 \sqrt{3} )}^{2}$

$y = 7+ 10 \sqrt{21} + 75$

$y = 82 + 10 \sqrt{21}$

$m_{h} = \frac{2}{ \frac{1}{x} + \frac{1}{y} }$

$m_{h} = \frac{2}{ \frac{1}{82 - 10 \sqrt{21} } + \frac{1}{82 + 10 \sqrt{21} } }$

$m_{h} = \frac{2}{ \frac{82 + 10 \sqrt{21} }{(82 - 10 \sqrt{21} )(82+ 10 \sqrt{21} )} + \frac{82 - 10 \sqrt{21} }{(82 - 10 \sqrt{21})(82 + 10 \sqrt{21} )} }$

$m_{h} = \frac{2}{ \frac{82 + 10 \sqrt{21} + 82 - 10 \sqrt{21} }{(82 - 10 \sqrt{21})(82 + 10 \sqrt{21} ) } }$

$m_{h} = \frac{2}{ \frac{164}{ {82}^{2} - {(10 \sqrt{21} )}^{2} } }$

$m_{h} = \frac{2}{ \frac{164}{6724 - 2100} }$

$m_{h} =\frac{2}{ \frac{164}{4624} }$

$m_{h} = 2 \times \frac{4624}{164}$

$m_{h} = \frac{4624}{82}$

$m_{h} = \frac{2312}{41}$