Question Number 116722 by ayenisamuel last updated on 06/Oct/20 $${find}\:{the}\:{range}\:{of}\:{values}\:{of}\:{k}\:{for} \\ $$$${which}\:{the}\:{equation}\:{e}^{{x}} −\mathrm{5}={k}\:{has}\:{no} \\ $$$${solution} \\ $$ Answered by Rio Michael last updated on 06/Oct/20…
Question Number 182247 by Shrinava last updated on 06/Dec/22 $$\mathrm{Find}: \\ $$$$\sqrt{\mathrm{9}\:+\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:−\:\sqrt{\mathrm{9}\:−\:\mathrm{4}\:\sqrt{\mathrm{5}}}\:=\:? \\ $$ Answered by mr W last updated on 06/Dec/22 $$\mathrm{9}\pm\mathrm{4}\sqrt{\mathrm{5}}=\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2}} \pm\mathrm{4}\sqrt{\mathrm{5}}+\mathrm{2}^{\mathrm{2}} =\left(\sqrt{\mathrm{5}}\pm\mathrm{2}\right)^{\mathrm{2}}…
Question Number 182241 by Matica last updated on 06/Dec/22 $$\:\mathrm{It}\:\mathrm{is}\:\mathrm{given}\:\mathrm{a}\:\mathrm{family}\:\mathrm{of}\:\mathrm{open}\:\mathrm{interval}\:\mathrm{set}\:\left({U}_{{r}} \right)_{{r}\in\mathbb{Q}} \:\mathrm{of}\:\mathbb{R} \\ $$$$\mathrm{that}\:\mathrm{satifies}\:\mathrm{condition}\:\forall{r}\in\mathbb{Q},\:{r}\in{U}_{{r}\:} . \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{family}\:\mathrm{set}\:\left({U}_{{r}} \right)_{{r}\in\mathbb{Q}} \mathrm{which}\:\mathrm{not}\:\mathrm{cover}\:\mathbb{R}\: \\ $$$$\mathrm{or}\:\forall\varepsilon>\mathrm{0},\:\:\lambda\left(\underset{{r}\in\mathbb{Q}} {\cup}\:{U}_{{r}} \:\right)\leqslant\:\varepsilon\:. \\ $$…
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Question Number 182216 by Acem last updated on 05/Dec/22 $$\:{Find}\:{a},\:{b},\:{c}\:\in\:\mathbb{N}\:;\:\mathrm{2}^{\:{a}} +\:\mathrm{4}^{\:{b}} +\:\mathrm{8}^{\:{c}} =\:\mathrm{328} \\ $$ Commented by SANOGO last updated on 06/Dec/22 $${thank}\:{you} \\ $$…
Question Number 116683 by mathocean1 last updated on 05/Oct/20 $$\mathrm{Given}\:\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}=\mathrm{16}\:\:\:\mathrm{we}\:\mathrm{know}\:\mathrm{that} \\ $$$$\mathrm{16}=\mathrm{4}^{\mathrm{2}} \:\:\mathrm{and}\:\mathrm{4}\:\mathrm{is}\:\mathrm{the}\:\mathrm{half}\:\mathrm{of}\:\mathrm{8}\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{successor}\:\mathrm{of}\:\mathrm{7}. \\ $$$$ \\ $$$$\mathrm{conjecture}\:\mathrm{the}\:\mathrm{result}\:\mathrm{of}\:\mathrm{this}\:\mathrm{sum}: \\ $$$$\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}+…+\mathrm{25} \\ $$ Answered by…
Question Number 182200 by Shrinava last updated on 05/Dec/22 $$\mathrm{1}.\:\:\:\underset{\boldsymbol{\mathrm{a}}\rightarrow−\boldsymbol{\mathrm{x}}} {\mathrm{lim}}\:\frac{\mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{a}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{a}^{\mathrm{3}} }\:=\:? \\ $$$$ \\ $$$$\mathrm{2}.\:\:\:\mathrm{cot}\:\mathrm{80}°\:\left(\mathrm{tan}\:\mathrm{10}°\:+\:\mathrm{2}\:\mathrm{tg}\:\mathrm{70}°\right)\:=\:? \\ $$ Answered by cortano1 last…
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Question Number 182188 by mathlove last updated on 05/Dec/22 $$\left(\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}+\mathrm{2}}+\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}−\mathrm{2}}\right)^{\mathrm{2014}} =? \\ $$ Answered by Frix last updated on 05/Dec/22 $$\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}}=\varphi\wedge\sqrt[{\mathrm{3}}]{−\mathrm{2}+\sqrt{\mathrm{5}}}=\frac{\mathrm{1}}{\varphi} \\ $$$$\varphi+\frac{\mathrm{1}}{\varphi}=\sqrt{\mathrm{5}} \\ $$$$\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2014}}…