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Question-191405

Question Number 191405 by Mingma last updated on 23/Apr/23 Commented by mr W last updated on 23/Apr/23 $${r}=\frac{{ab}}{\:\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }+\left(\mathrm{1}+\sqrt{\mathrm{3}}\right){a}+\mathrm{2}{b}} \\ $$$${with}\:{a}=\mathrm{3},\:{b}=\mathrm{4}: \\ $$$${r}=\frac{\mathrm{12}}{\:\mathrm{16}+\mathrm{3}\sqrt{\mathrm{3}}}\approx\mathrm{0}.\mathrm{566} \\…

Question-191404

Question Number 191404 by Mingma last updated on 23/Apr/23 Answered by mr W last updated on 23/Apr/23 $${as}\:{Q}\mathrm{191305}\:{but}\:\theta=\mathrm{0}° \\ $$$${r}=\frac{{ab}}{{a}+{b}+\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }+\left({a}+\sqrt{\mathrm{3}}{b}\right)\mathrm{sin}\:\theta+\left(\sqrt{\mathrm{3}}{a}+{b}\right)\mathrm{cos}\:\theta} \\ $$$${with}\:\theta=\mathrm{0}° \\…

Question-125868

Question Number 125868 by I want to learn more last updated on 14/Dec/20 Answered by mindispower last updated on 14/Dec/20 $$\frac{\mathrm{1}}{\mathrm{2}{x}}+\frac{\mathrm{1}}{\mathrm{2}{y}}\geqslant\frac{\mathrm{2}}{{x}+{y}}.\Leftrightarrow\left({x}+{y}\right)^{\mathrm{2}} \geqslant\mathrm{4}{xy},{x},{y}>\mathrm{0} \\ $$$$\Leftrightarrow\left({x}−{y}\right)^{\mathrm{2}} \geqslant\mathrm{0}\:\:{true}…

Question-191403

Question Number 191403 by Mingma last updated on 23/Apr/23 Answered by mr W last updated on 23/Apr/23 $${as}\:{Q}\mathrm{191305}\:{but}\:\theta=\mathrm{45}° \\ $$$${r}=\frac{{ab}}{{a}+{b}+\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }+\left({a}+\sqrt{\mathrm{3}}{b}\right)\mathrm{sin}\:\theta+\left(\sqrt{\mathrm{3}}{a}+{b}\right)\mathrm{cos}\:\theta} \\ $$$${with}\:\theta=\mathrm{45}° \\…

1-1-1-2-1-4-2-1-2-2-2-4-3-1-3-2-3-4-50-1-50-2-50-4-

Question Number 125864 by MathSh last updated on 14/Dec/20 $$\frac{\mathrm{1}}{\mathrm{1}+\mathrm{1}^{\mathrm{2}} +\mathrm{1}^{\mathrm{4}} }+\frac{\mathrm{2}}{\mathrm{1}+\mathrm{2}^{\mathrm{2}} +\mathrm{2}^{\mathrm{4}} }+\frac{\mathrm{3}}{\mathrm{1}+\mathrm{3}^{\mathrm{2}} +\mathrm{3}^{\mathrm{4}} }+…+\frac{\mathrm{50}}{\mathrm{1}+\mathrm{50}^{\mathrm{2}} +\mathrm{50}^{\mathrm{4}} }=? \\ $$ Answered by Dwaipayan Shikari last…

Question-125862

Question Number 125862 by mathdave last updated on 14/Dec/20 Answered by bramlexs22 last updated on 14/Dec/20 $$\sqrt{\frac{\mathrm{7}^{\mathrm{2012}} \left(\mathrm{49}−\mathrm{1}\right)}{\mathrm{12}}}\:=\:\sqrt{\mathrm{7}^{\mathrm{2012}} ×\mathrm{4}} \\ $$$$\:=\:\mathrm{2}×\mathrm{7}^{\mathrm{1006}} \:=\:{a}×\mathrm{7}^{{b}} \\ $$$${a}=\mathrm{2}\:\wedge\:{b}=\mathrm{1006} \\…