Question

Exercițiul 13 și 14 va rog rapid

Answer 1

$\displaystyle 13.~~\frac{a+b+c}{3} =7\sqrt{3}\\ \\ a=-4+6\sqrt{3};~b=14\sqrt{3}  \\ \\ \frac{a+b+c}{3} =7\sqrt{3} \Rightarrow \frac{-4+6\sqrt{3}+14\sqrt{3}+c}{3} =7\sqrt{3} \Rightarrow \\ \\ \Rightarrow -4+20\sqrt{3} +c=7\sqrt{3} \cdot 3 \Rightarrow-4+20\sqrt{3}+c=21\sqrt{3} \Rightarrow\\ \\ \Rightarrow c=21\sqrt{3} -20\sqrt{3}+4\Rightarrow \mathbf{c=\sqrt{3} +4}\\ \\M_a=\frac{a+b}{2} \Rightarrow M_a=\frac{-4+6\sqrt{3}+14\sqrt{3}}{2} \Rightarrow M_a=\frac{20\sqrt{3}-4}{2}\Rightarrow$

$\displaystyle \Rightarrow M_a=\frac{\not2(10\sqrt{3}-2)}{\not2}\Rightarrow M_a=\mathbf{10\sqrt{3} -2}$

$\displaystyle 14.~~a)\sqrt{14};~\sqrt{56}\\ \\ ~M_g=\sqrt{\sqrt{14}\cdot\sqrt{56}}\Rightarrow M_g=\sqrt{\sqrt{784}}\Rightarrow M_g=\sqrt{28} \Rightarrow \mathbf{M_g=2\sqrt{7} }\\ \\ \\ \\b)~13-\sqrt{69};~13+\sqrt{69}\\\\ M_g=\sqrt{(13-\sqrt{69})(13+\sqrt{69})}\Rightarrow M_g=\sqrt{13^2-(\sqrt{69})^2}\Rightarrow\\\\\Rightarrow M_g=\sqrt{169-69} \Rightarrow M_g=\sqrt{100}\Rightarrow \mathbf{M_g=10}$

$\displaystyle c)~\frac{10\sqrt{2}-\sqrt{7}}{8};~\frac{\sqrt{200}+\sqrt{7}}{2}\\\\ \frac{\sqrt{200}+\sqrt{7}}{2}=\frac{10\sqrt{2}+\sqrt{7}}{2}\\\\ M_g=\sqrt{\frac{10\sqrt{2}-\sqrt{7}}{8}\cdot\frac{10\sqrt{2}+\sqrt{7}}{2}}\Rightarrow M_g=\sqrt{\frac{(10\sqrt{2}-\sqrt{7})(10\sqrt{2}+\sqrt{7})}{16}}\Rightarrow\\\\\Rightarrow M_g=\sqrt{\frac{(10\sqrt{2})^2-(\sqrt{7})^2}{16}}\RightarrowM_g=\sqrt{\frac{200-7}{16}}\Rightarrow M_g=\sqrt{\frac{193}{16}}\Rightarrow$

$\displaystyle \Rightarrow M_g=\frac{\sqrt{193}}{\sqrt{16}}\Rightarrow \mathbf{M_g=\frac{\sqrt{193}}{4}}$

$\displaystyle d)~5\sqrt{2}+3\sqrt{18}-2\sqrt{50};~\frac{6\sqrt{2}}{3}+\frac{\sqrt{98}}{5}\\\\5\sqrt{2}+3\sqrt{18}-2\sqrt{50}=5\sqrt{2}+3\cdot3\sqrt{2}-2\cdot 5\sqrt{2}=5\sqrt{2}+9\sqrt{2}-10\sqrt{2}=\\\\=4\sqrt{2}\\\\\frac{6\sqrt{2}}{3}+\frac{\sqrt{98}}{5}=2\sqrt{2}+\frac{7\sqrt{2}}{5}=\frac{10\sqrt{2}+7\sqrt{2}}{5}=\frac{17\sqrt{2}}{5}$

$\displaystyle M_g=\sqrt{4\sqrt{2}\cdot\frac{17\sqrt{2}}{5}}\Rightarrow M_g=\sqrt{\frac{136}{5}}\Rightarrow M_g=\frac{\sqrt{136}}{\sqrt{5}}\Rightarrow M_g=\frac{\sqrt{680}}{5}\Rightarrow \\ \\ \Rightarrow \mathbf{M_g=\frac{2\sqrt{170}}{5}}$