Question

1. calculati media aritmetica si media geometrica a numerelor :
a) $3 \sqrt{2$ si $\frac{1}{2 \sqrt{2}}$]
b) $\sqrt{20}$ si $\sqrt{80$
c) $\sqrt{5}+1 si \sqrt{5$-1

Answer 1

$a) \\ m_a= \frac{3 \sqrt{2}+ \frac{1}{2\sqrt{2}} }{2}= \frac{3 \sqrt{2}*2\sqrt{2}+1}{2*2\sqrt{2}}= \frac{6*2+1}{4\sqrt{2}}= \frac{13}{4\sqrt{2}} = \frac{13\sqrt{2}}{4*2} =\boxed{\frac{13\sqrt{2}}{8}} \\ m_g= \sqrt{3\sqrt{2}*\frac{1}{2\sqrt{2}}}=\sqrt{\frac{3\sqrt{2}}{2\sqrt{2}}}= \boxed{\sqrt{\frac{3}{2}}}$

$b) \\ m_a= \frac{ \sqrt{20}+ \sqrt{80}}{2}=\frac{ 2\sqrt{5}+ 4\sqrt{5}}{2}= \frac{ 6\sqrt{5}}{2}=\boxed{3\sqrt{5}} \\ m_g= \sqrt{ \sqrt{20}* \sqrt{80}}= \sqrt{ \sqrt{20*80}}= \sqrt{ \sqrt{1600}}= \sqrt{40} =\boxed{2 \sqrt{10} }$

$c) \\ m_a= \frac{ \sqrt{5}+1 +\sqrt{5}-1 }{2}= \frac{ \sqrt{5}+\sqrt{5} }{2}= \frac{ 2\sqrt{5}}{2}=\boxed{\sqrt{5}} \\ m_g = \sqrt{(\sqrt{5}+1)(\sqrt{5}-1)}= \sqrt{(\sqrt{5})^2-(1)^2}= \sqrt{5-1}= \sqrt{4}=\boxed{2}$