Question Number 145279 by mathdanisur last updated on 03/Jul/21 $$\mathrm{2}^{\boldsymbol{{a}}!} \:+\:\mathrm{2}^{\boldsymbol{{b}}!} \:+\:\mathrm{2}^{\boldsymbol{{c}}!} \:=\:\boldsymbol{{x}} \\ $$$${Find}\:{natural}\:{numbers}\:\boldsymbol{{a}};\boldsymbol{{b}};\boldsymbol{{c}}\:{such} \\ $$$${that}\:{the}\:{number}\:“\boldsymbol{{x}}''\:{is}\:{a}\:{cube}\:{of} \\ $$$${any}\:{number}. \\ $$ Answered by SinNombre last…
Question Number 145259 by mathdanisur last updated on 03/Jul/21 $$\int\:\frac{\left(\mathrm{3}\sqrt{{x}}+\mathrm{2}\right)^{\mathrm{5}} }{\:\sqrt{{x}}}\:{dx}\:=\:? \\ $$ Answered by Dwaipayan Shikari last updated on 03/Jul/21 $${put}\:\sqrt{{x}}={t}\Rightarrow\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}{dx}={dt} \\ $$$$=\mathrm{2}.\mathrm{3}^{\mathrm{5}} \int\left({t}+\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{5}}…
Question Number 145240 by loveineq last updated on 03/Jul/21 $$\mathrm{Let}\:{a},{b},{c}\:\geqslant\:\mathrm{0}\:\mathrm{and}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{cyc}} {\sum}{a}^{\mathrm{3}} +\underset{{cyc}} {\sum}\left({a}+{b}\right)^{\mathrm{3}} \:\leqslant\:\mathrm{27} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} \:\geqslant\:\frac{\mathrm{1}}{\mathrm{2}}\left[{c}^{\mathrm{3}}…
Question Number 145238 by liberty last updated on 03/Jul/21 Answered by gsk2684 last updated on 03/Jul/21 $$\mathrm{x}=\mathrm{1}\:\mathrm{or}\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$ Answered by Rasheed.Sindhi last updated on…
Question Number 145231 by mathdanisur last updated on 03/Jul/21 $$\mathrm{1}<{a}\leqslant{b}\:\:{then}\:{find} \\ $$$$\underset{\:\boldsymbol{{a}}} {\overset{\:\boldsymbol{{b}}} {\int}}\:{tan}^{-\mathrm{1}} \left(\frac{\mathrm{3}{x}}{\mathrm{1}-\mathrm{2}{x}^{\mathrm{2}} }\right){dx}=? \\ $$ Commented by Dwaipayan Shikari last updated on…
Question Number 145224 by mathdanisur last updated on 03/Jul/21 $$\sqrt{\mathrm{2}\sqrt{\mathrm{3}}\:+\:\mathrm{2}}\:-\:\sqrt{\sqrt{\mathrm{3}}\:-\:\sqrt{\mathrm{2}}}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 03/Jul/21 $$\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{2}}+\mathrm{1}\right)^{\mathrm{2}} \:=\:\mathrm{6}+\mathrm{2}\sqrt{\mathrm{6}}+\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{2}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$${x}\:=\:\sqrt{\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{2}}−\sqrt{\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}}}…
Question Number 14157 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 29/May/17 Commented by ajfour last updated on 30/May/17 $${kindly}\:{view}\:{my}\:{solution}\:{in} \\ $$$${Q}.\mathrm{14306}\:{solved}\:{by}\:{coordinate} \\ $$$${geometry}.. \\ $$ Commented by…
Question Number 14147 by RasheedSindhi last updated on 28/May/17 $$\mathrm{Determine}: \\ $$$$\left({a}\right)\:\mathrm{e}^{\omega} \\ $$$$\left({b}\right)\:\omega^{\mathrm{e}} \\ $$ Commented by prakash jain last updated on 28/May/17 $${e}^{{w}}…
Question Number 14148 by RasheedSindhi last updated on 28/May/17 $$\mathrm{Determine} \\ $$$$\left({a}\right)\:{i}^{{i}} \\ $$$$\left({b}\right)\:\omega^{\omega} \\ $$$$\left({c}\right)\:{i}^{\omega} \\ $$$$\left({d}\right)\:\omega^{{i}} \\ $$ Answered by prakash jain last…
Question Number 145198 by loveineq last updated on 03/Jul/21 $$\mathrm{Let}\:{a}\geqslant{b}\geqslant{c}\geqslant\mathrm{0}\:,\:{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} \neq\mathrm{0}\:\mathrm{and}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} \:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\leqslant\:\frac{{a}^{\mathrm{3}} +\left({b}+{c}\right)^{\mathrm{3}} +{b}^{\mathrm{3}} +\left({c}+{a}\right)^{\mathrm{3}} }{{c}^{\mathrm{3}} +\left({a}+{b}\right)^{\mathrm{3}} }\:\leqslant\:\mathrm{2} \\ $$$$\mathrm{Determine}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}.…