Question Number 207385 by hardmath last updated on 13/May/24 $$\mathrm{Find}: \\ $$$$\sqrt{\left(\mathrm{2},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{2}} }\:−\:\sqrt[{\mathrm{3}}]{\left.\left(\mathrm{1},\mathrm{5}−\sqrt{\mathrm{5}}\right)^{\mathrm{3}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} }\:−\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\frac{\mathrm{7}\pi}{\mathrm{4}} \\ $$ Commented by Frix last updated on 13/May/24 $$=\frac{\mathrm{7}}{\mathrm{2}}−\sqrt{\mathrm{5}}−\frac{\sqrt{\mathrm{6}\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{5}}\right)}}{\mathrm{4}}−\frac{\sqrt{\mathrm{2}\left(−\mathrm{3}+\mathrm{2}\sqrt{\mathrm{5}}\right.}}{\mathrm{4}}\mathrm{i}…
Question Number 207387 by sniper237 last updated on 13/May/24 $${Let}\:\:{cardE}={n}\:,\:{and}\:\:{the}\:{set}\:{of}\:{parts} \\ $$$${S}=\left\{\left({A},{B}\right)\in{P}\left({E}\right)×{P}\left({E}\right)\:/\:\:{A}\cap{B}=\varnothing\right\} \\ $$$${Show}\:{that}\:\:{cardS}=\:\mathrm{3}^{{n}} \\ $$ Answered by Berbere last updated on 13/May/24 $${if}\:{card}\left({A}\right)={k};{E}={A}\cup\overset{−} {{A}}\:\:{the}\:{number}\:{of}\:{subset}\:{of}\:{card}={k}…
Question Number 207381 by efronzo1 last updated on 13/May/24 Answered by mr W last updated on 13/May/24 $$\frac{{BD}\centerdot{CE}}{{AC}\centerdot{DE}} \\ $$$$=\frac{\left({BE}+{DE}\right)\centerdot{CE}}{\left({AE}+{CE}\right)\centerdot{DE}} \\ $$$$=\frac{\left(\frac{{BE}}{{DE}}+\mathrm{1}\right)\centerdot{CE}}{\left(\frac{{AE}}{{CE}}+\mathrm{1}\right)\centerdot{DE}} \\ $$$$=\frac{\frac{{BC}}{{AD}}+\mathrm{1}}{\frac{{BC}}{{AD}}+\mathrm{1}} \\…
Question Number 207382 by efronzo1 last updated on 13/May/24 Answered by sniper237 last updated on 13/May/24 $$\overset{{X}=^{\mathrm{3}} \sqrt{{x}−\mathrm{2}}} {=}\underset{{X}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{X}^{\mathrm{6}} +\mathrm{2}{X}^{\mathrm{3}} +{X}}{\:^{\mathrm{3}} \sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}{X}^{\mathrm{3}} +\mathrm{4}}−{X}^{\mathrm{3}} \sqrt{\mathrm{3}{X}^{\mathrm{3}}…
Question Number 207383 by Shrodinger last updated on 13/May/24 $$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$ Commented by Frix last updated on 13/May/24 $$\mathrm{Impossible}. \\ $$ Commented…
Question Number 207408 by hardmath last updated on 13/May/24 $$\boldsymbol{\mathrm{z}}\:\:=\:\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:−\:\frac{\mathrm{1}}{\mathrm{2}}\:\boldsymbol{\mathrm{i}}\:\:\:\:\:\mathrm{find}:\:\:\boldsymbol{\mathrm{z}}^{\mathrm{11}} \:=\:? \\ $$ Answered by Frix last updated on 13/May/24 $${z}=\mathrm{e}^{−\frac{\pi}{\mathrm{6}}\mathrm{i}} \\ $$$${z}^{\mathrm{11}} =\mathrm{e}^{−\frac{\mathrm{11}\pi}{\mathrm{6}}\mathrm{i}} =\mathrm{e}^{\left(\mathrm{2}\pi−\frac{\mathrm{11}\pi}{\mathrm{6}}\right)\mathrm{i}}…
Question Number 207374 by sniper237 last updated on 13/May/24 $${Show}\:{that}\:\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} ={C}_{\mathrm{2}{n}} ^{{n}} \\ $$ Answered by mr W last updated on…
Question Number 207407 by hardmath last updated on 13/May/24 Commented by Frix last updated on 13/May/24 $$\mathrm{2}.\:…=\mathrm{2}\sqrt{\mathrm{3}}\mathrm{cos}\:\mathrm{10}° \\ $$ Commented by Frix last updated on…
Question Number 207402 by MATHEMATICSAM last updated on 13/May/24 $${f}\left({x}\right)\:+\:\mathrm{2}{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{3}{x}. \\ $$$$\:{f}\:'\left({x}\right)\:=\:? \\ $$ Answered by Frix last updated on 13/May/24 $${x}={t}:\:{f}\left({t}\right)+\mathrm{2}{f}\left(\frac{\mathrm{1}}{{t}}\right)=\mathrm{3}{t} \\ $$$${x}=\frac{\mathrm{1}}{{t}}:\:{f}\left(\frac{\mathrm{1}}{{t}}\right)+\mathrm{2}{f}\left({t}\right)=\frac{\mathrm{3}}{{t}} \\…
Question Number 207394 by hardmath last updated on 13/May/24 $$\frac{\mathrm{a}}{\mathrm{b}}\:\:=\:\:\frac{\mathrm{c}}{\mathrm{d}} \\ $$$$\mathrm{a}^{\mathrm{3}} \:−\:\mathrm{b}^{\mathrm{3}} \:=\:\mathrm{625} \\ $$$$\mathrm{c}^{\mathrm{3}} \:−\:\mathrm{d}^{\mathrm{3}} \:=\:\mathrm{1} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:=\:? \\ $$ Answered by mr…