Question

10. Calculaţi media aritmetică şi media geometrică a numerelor: a) √3 şi 3√3
b) 2√125 şi 7√80 c) √24 și 3/√150 d) 3,2 şi 4,8 e) 1,6√3 şi 5√27 f) 5√26 și 2√104​

Answer 1

Răspuns:

$a)ma = \frac{ \sqrt{3} + 3 \sqrt{3} }{2} = \frac{4 \sqrt{3} }{2} = 2 \sqrt{3} \\ mg = \sqrt{ \sqrt{3} \times 3 \sqrt{3} } = \sqrt{3 \times 3 } = \sqrt{9} = 3$

$b)2 \sqrt{125} = 2 \sqrt{ {5}^{2} \times 5} = 2 \times 5 \times \sqrt{5} = 10 \sqrt{5} \\ 7 \sqrt{80} = 7 \times \sqrt{16 \times 5} = 7 \times 4 \sqrt{5} = 28 \sqrt{5} \\ ma = \frac{10 \sqrt{5} + 28 \sqrt{5} }{2} = \frac{38 \sqrt{5} }{2} = 19 \sqrt{5} \\ mg = \sqrt{10 \sqrt{5} \times 28 \sqrt{5} } = \sqrt{280 \times 5} = \sqrt{1440} = \sqrt{144} \times \sqrt{10} = 12 \sqrt{10}$

$c)ma = \frac{ \sqrt{24} + 3 \sqrt{150} }{2} = \frac{ \sqrt{4 \times 6} + 3 \sqrt{25 \times 6} }{2} = \frac{2 \sqrt{6} + 3 \times 5 \sqrt{6} }{2} = \frac{2 \sqrt{6} + 15 \sqrt{6} }{2} = \frac{17 \sqrt{6} }{2} \\ mg = \sqrt{ \sqrt{24} \times \sqrt{150} } = \sqrt{2 \sqrt{6} \times 5 \sqrt{6} } = \sqrt{10 \times 6} = \sqrt{60} = \sqrt{4 \times 15} = 2 \sqrt{15}$

$d)ma = \frac{3.2 + 4.8}{2} = \frac{8}{2} = 4 \\ mg = \sqrt{3.2 \times 4.8} = \sqrt{ \frac{32}{10} \times \frac{48}{10} } = \sqrt{ \frac{16}{5} \times \frac{24}{5} } = \sqrt{ \frac{16}{5} } \times \sqrt{ \frac{24}{5} } = \frac{4}{ \sqrt{5} } \times \frac{2 \sqrt{6} }{ \sqrt{5} } = \frac{8 \sqrt{6} }{5}$

$e)ma = \frac{1.6 \sqrt{3} + 5 \sqrt{27} }{2} = \frac{1.6 \sqrt{3} + 5 \sqrt{9 \times 3} }{2} = \frac{1.6 \sqrt{3} + 5 \times 3 \sqrt{3} }{2} = \frac{1.6 \sqrt{3} + 15 \sqrt{3} }{2} = \frac{16.6 \sqrt{3} }{2} = 8.3 \sqrt{2} \\ mg = \sqrt{1.6 \sqrt{3} \times 5 \sqrt{27} } = \sqrt{8 \times \sqrt{81} } = \sqrt{8 \times 9} = \sqrt{8} \times \sqrt{9} = \sqrt{4 \times 2} \times 3 = 2 \sqrt{2} \times 3 = 6 \sqrt{2}$

$f)ma = \frac{5 \sqrt{26} + 2 \sqrt{104} }{2} = \frac{5 \sqrt{26} + 2 \sqrt{26 \times 4} }{2} = \frac{5 \sqrt{26} + 2 \times 2 \sqrt{26} }{2} = \frac{5 \sqrt{26} + 4 \sqrt{26} }{2} = \frac{9 \sqrt{26} }{2} \\ mg = \sqrt{5 \sqrt{26} \times 2 \sqrt{104} } = \sqrt{5 \sqrt{26} \times 4 \sqrt{26} } = \sqrt{20 \times 26} = \sqrt{20} \times \sqrt{26} = \sqrt{4 \times 5} \times \sqrt{26} = 2 \sqrt{5} \times \sqrt{26} = 2 \sqrt{130}$