Category: Algebra

Question Number 145393 by physicstutes last updated on 04/Jul/21 $$\mathrm{The}\:\mathrm{number}\:\frac{\mathrm{2}}{\mathrm{13}}\:\mathrm{expressed}\:\mathrm{as}\:\mathrm{a}\:\mathrm{decimal}\:\mathrm{is}\:\mathrm{0}.\mathrm{153846153846}… \\ $$$$\mathrm{The}\:\mathrm{200th}\:\mathrm{and}\:\mathrm{300th}\:\mathrm{digits}\:\mathrm{are}? \\ $$ Answered by mr W last updated on 04/Jul/21 $${mod}\:\left(\mathrm{200},\mathrm{6}\right)=\mathrm{2} \\ $$$$\Rightarrow\mathrm{200}{th}\:{digit}\:{is}\:\mathrm{5}…

Question Number 145383 by mathdanisur last updated on 04/Jul/21 $${if}\:\:{a};{b};{c}\in\mathbb{R}^{+} \\ $$$${find}\:\:\left(\sqrt[{\mathrm{3}}]{{abc}}\:+\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{b}}\:+\:\frac{\mathrm{1}}{\mathrm{4}{c}}\right)_{\boldsymbol{{min}}} =\:? \\ $$ Answered by mnjuly1970 last updated on 04/Jul/21 $$\:\mathrm{A}\:\geqslant\:\sqrt[{\mathrm{3}}]{{abc}}\:+\mathrm{3}\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{8}{abc}}} \\ $$$$\:\:\:\:\:\:=\sqrt[{\mathrm{3}}]{{abc}}\:+\frac{\mathrm{3}}{\mathrm{2}}\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{{abc}}}\:\overset{{am}−{gm}}…

Question Number 145344 by mathdanisur last updated on 04/Jul/21 $$\boldsymbol{{z}}^{\mathrm{4}} \:=\:\mathrm{49}\:-\:\mathrm{20}\sqrt{\mathrm{6}}\:\:\Rightarrow\:\boldsymbol{{z}}=? \\ $$ Answered by liberty last updated on 04/Jul/21 $$\:{z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{49}−\mathrm{2}\sqrt{\mathrm{600}}}\:=\sqrt{\left(\mathrm{25}+\mathrm{24}\right)−\mathrm{2}\sqrt{\mathrm{25}×\mathrm{24}}} \\ $$$${z}^{\mathrm{2}} \:=\:\sqrt{\mathrm{25}}−\sqrt{\mathrm{24}}\:=\:\mathrm{5}−\mathrm{2}\sqrt{\mathrm{6}}\:…

Question Number 145337 by physicstutes last updated on 04/Jul/21 $$\mathrm{The}\:\mathrm{Area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},{A}\left({t}\right),\mathrm{is}\:\mathrm{increased}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate} \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{its}\:\mathrm{perimeter}\:,{A}\left({t}\right)\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\frac{{dA}}{{dt}}= \\ $$$$\mathrm{A}.\:\mathrm{4}{A}\:\:\:\:\:\:\:\:\:\mathrm{B}.\:\mathrm{2}{A}\:\:\:\:\:\:\:\:\:\mathrm{C}.−\mathrm{4}\sqrt{{A}}\:\:\:\:\:\:\:\:\mathrm{D}.\:\mathrm{4}\sqrt{{A}} \\ $$ Answered by gsk2684 last updated on 06/Jul/21…

Question Number 145324 by mathdanisur last updated on 04/Jul/21 $$\int\:\frac{{dx}}{\mathrm{1}\:+\:{x}^{\mathrm{6}} }\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 04/Jul/21 $$\frac{\mathrm{1}}{\mathrm{6}}\mathrm{atan}\left(\frac{{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\right)+\frac{\mathrm{1}}{\mathrm{3}}\mathrm{atan}{x}+\frac{\sqrt{\mathrm{3}}}{\mathrm{12}}\mathrm{ln}\mid\frac{{x}^{\mathrm{2}} +\sqrt{\mathrm{3}}{x}+\mathrm{1}}{{x}^{\mathrm{2}} −\sqrt{\mathrm{3}}{x}+\mathrm{1}}\mid+\mathrm{C} \\…