Question Number 204658 by hardmath last updated on 24/Feb/24 $$\mathrm{If}\:\:\:\mathrm{a}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{3}}\:+\:\mathrm{1} \\ $$$$\mathrm{Find}\:\:\:\left(\frac{\mathrm{4}\:−\:\mathrm{a}}{\mathrm{a}}\right)^{\mathrm{6}} =\:? \\ $$ Answered by Rasheed.Sindhi last updated on 24/Feb/24 $$\:\mathrm{a}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{3}}\:+\:\mathrm{1};\:\left(\frac{\mathrm{4}\:−\:\mathrm{a}}{\mathrm{a}}\right)^{\mathrm{6}} =\:? \\…
Question Number 204642 by BaliramKumar last updated on 24/Feb/24 $$\mathrm{If}\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\:\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }\:+\:………….\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{then}\:\:\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }\:+\:………….\:=\:? \\ $$$$ \\ $$…
Question Number 204632 by mr W last updated on 23/Feb/24 Answered by witcher3 last updated on 23/Feb/24 $$\left.\mathrm{x}>\mathrm{1};\mathrm{x}=\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{t}\right)};\mathrm{t}\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\right. \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{t}\right)}+\frac{\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{t}\right)}}{\:\sqrt{\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{t}\right)}−\mathrm{1}}}=\frac{\mathrm{1}}{\mathrm{sin}\left(\mathrm{t}\right)}+\frac{\mathrm{1}}{\mathrm{cos}\left(\mathrm{t}\right)}=\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$\mathrm{cauchy}\:\mathrm{shwartz}\:\left(\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{sin}\left(\mathrm{t}\right)}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{cos}\left(\mathrm{t}\right)}}\right)^{\mathrm{2}} \right)\left(\left(\sqrt{\mathrm{sin}\left(\mathrm{t}\right)}\right)^{\mathrm{2}}…
Question Number 204603 by liuxinnan last updated on 23/Feb/24 Commented by TonyCWX08 last updated on 23/Feb/24 $${Please}\:{use}\:{the}\:{latex}\:{form}. \\ $$$${I}\:{don}'{t}\:{understand}\:{what}\:{you}'{re}\:{trying}\:{to}\:{express}… \\ $$ Commented by liuxinnan last…
Question Number 204628 by MM42 last updated on 23/Feb/24 $$ \\ $$ Answered by MM42 last updated on 23/Feb/24 Answered by A5T last updated on…
Question Number 204621 by hardmath last updated on 23/Feb/24 $$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{If}\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$ Answered by A5T last updated on 23/Feb/24 $$\frac{{a}+{b}+{c}}{\mathrm{3}}\geqslant\left(\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{9}}\Rightarrow{a}+{b}+{c}\geqslant\frac{\mathrm{1}}{\mathrm{3}}…
Question Number 204617 by Abdullahrussell last updated on 23/Feb/24 Commented by Frix last updated on 23/Feb/24 $$\mathrm{33} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 204618 by es last updated on 23/Feb/24 $${if}\:\:\mathrm{7}{x}=\frac{\pi}{\mathrm{2}}\rightarrow\frac{{cosxsin}\mathrm{2}{xtan}\mathrm{3}{x}}{{cot}\mathrm{4}{xcos}\mathrm{5}{xsin}\mathrm{6}{x}}=? \\ $$ Answered by A5T last updated on 23/Feb/24 $$\frac{\frac{{cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}}{\frac{{cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}{{sin}\left(\mathrm{4}{x}\right)}} \\ $$$$=\frac{{sin}\left(\mathrm{4}{x}\right){cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right){cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}=\mathrm{1} \\ $$$$\left[{since}\:{sin}\left(\mathrm{4}{x}\right)={cos}\left(\mathrm{3}{x}\right);{cos}\left({x}\right)={sin}\left(\mathrm{6}{x}\right);\right. \\…
Question Number 204615 by lepuissantcedricjunior last updated on 26/Feb/24 $$\:\:\:\:\:\:\:\:\frac{\boldsymbol{\mathrm{exercice}}\:}{} \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{prouver}}\:\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\boldsymbol{\mathrm{x}}} \boldsymbol{{sin}}\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\pi}}\right)\boldsymbol{{d}\mathrm{x}{d}\mathrm{y}}=\boldsymbol{\pi} \\ $$$$\: \\ $$$$\:\:……………\boldsymbol{{prof}}\:\boldsymbol{{cedric}}\:\boldsymbol{{junior}}……….. \\ $$$$ \\ $$…
Question Number 204610 by mnjuly1970 last updated on 23/Feb/24 $$ \\ $$$$\:\:\:{If}\:,\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\:\mathrm{2}^{\mathrm{2}{x}} −\:{log}_{\mathrm{3}} \:\left(\:{x}+\mathrm{3}\:\right)\:\:\:\:;\:\:\:{x}\:\geqslant\mathrm{5}}\\{\:{f}\:\left(\mathrm{1}+\:{x}\:\right)\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:;\:\:{x}\:<\:\mathrm{5}}\end{cases}\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{f}\:\left(\mathrm{0}\:\right)=\:? \\ $$$$ \\ $$ Answered by Rasheed.Sindhi…