Question

Calculati media aritmetica si media geometrică a numerelor
$7 - 4 \sqrt{3} si 7 + 4 \sqrt{3}$
$8 - \sqrt{39s i \: } 8 - \sqrt{39}$
$7 - 2 \sqrt{6} si \: 7 + 2 \sqrt{6}$
$6 - 2 \sqrt{5}si6 - 2 \sqrt{5}$
​

Answer 1

$1)m_{a} = \frac{7 - 4 \sqrt{3} + 7 + 4 \sqrt{3} }{2} = \frac{14}{2} = 7$

$m_{g} = \sqrt{(7 - 4 \sqrt{3})(7 + 4 \sqrt{3} ) }$

$m_{g} = \sqrt{ {7}^{2} - {(4 \sqrt{3}) }^{2} }$

$m_{g} = \sqrt{49 - 48}$

$m_{g} = \sqrt{1}$

$m_{g} = 1$

$2)m_{a} = \frac{8 - \sqrt{39} + 8 + \sqrt{39} }{2} = \frac{16}{2} = 8$

$m_{g} = \sqrt{(8 - \sqrt{39})(8 + \sqrt{39} ) }$

$m_{g} = \sqrt{ {8}^{2} - {( \sqrt{39}) }^{2} }$

$m_{g} = \sqrt{64 - 39}$

$m_{g} = \sqrt{15}$

$3)m_{a} = \frac{7 - 2 \sqrt{6} + 7 + 2 \sqrt{6} }{2} = \frac{14}{2} = 7$

$m_{g} = \sqrt{(7 - 2 \sqrt{6})(7 + 2 \sqrt{6} ) }$

$m_{g} = \sqrt{ {7}^{2} - {(2 \sqrt{6}) }^{2} }$

$m_{g} = \sqrt{49 - 24}$

$m_{g} = \sqrt{25}$

$m_{g} = 5$

$4)m_{a} = \frac{ 6- 2 \sqrt{5} + 6 + 2 \sqrt{5} }{2} = \frac{12}{2} = 6$

$m_{g} = \sqrt{(6 - 2 \sqrt{5})(6 + 2 \sqrt{5}) }$

$m_{g} = \sqrt{ {6}^{2} - {(2 \sqrt{5} )}^{2} }$

$m_{g} = \sqrt{36 - 20}$

$m_{g} = \sqrt{16}$

$m_{g} = 4$