Question Number 192542 by peter frank last updated on 20/May/23 Answered by Frix last updated on 20/May/23 $$\sqrt{\mathrm{i}}=\sqrt{\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} }=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} =\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{i} \\ $$$$\left(\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} \right)^{\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{i}} =\:\:\:\:\:\:\:\:\:\:\left[\left({a}^{{b}} \right)^{{c}}…
Question Number 126990 by physicstutes last updated on 26/Dec/20 $$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\: \\ $$$$\:{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{{n}} {\int}}\left[{x}\right]\:{dx}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\:\mathrm{where}\:\left[{x}\right]\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}\:\mathrm{of}\:{x}. \\ $$ Answered by mr W last updated…
Question Number 126989 by arash sharifi last updated on 25/Dec/20 $$\int{e}^{\sqrt{{x}}} {dx} \\ $$ Answered by bramlexs22 last updated on 26/Dec/20 $${let}\:\sqrt{{x}}\:=\:{u}\:\Rightarrow{x}\:=\:{u}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:{dx}\:=\:\mathrm{2}{u}\:{du}\: \\…
Question Number 192508 by Spillover last updated on 19/May/23 Commented by Frix last updated on 19/May/23 $$\mathrm{They}\:\mathrm{solved}\:\mathrm{a}\:\mathrm{similar}\:\mathrm{question}\:\mathrm{for}\:\mathrm{you}. \\ $$$$\mathrm{Are}\:\mathrm{you}\:\mathrm{too}\:\mathrm{stupid}\:\mathrm{or}\:\mathrm{simply}\:\mathrm{too}\:\mathrm{lazy}? \\ $$ Answered by Spillover last…
Question Number 126966 by ajfour last updated on 25/Dec/20 Commented by ajfour last updated on 25/Dec/20 $${With}\:{the}\:{help}\:{of}\:{the}\:{intersection} \\ $$$${of}\:{the}\:{inclined}\:{parabola},\:{find}\: \\ $$$${a}\:{root}\:{of}\:{the}\:{cubic}:\:\:\boldsymbol{{y}}=\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{{x}}−\boldsymbol{{c}}. \\ $$$${Assume}\:{shape}\:{of}\:{parabola}\:{the} \\…
Question Number 126950 by AST last updated on 05/Dec/22 Answered by floor(10²Eta[1]) last updated on 25/Dec/20 $$\mathrm{A}.\mathrm{1}+\mathrm{2}^{\mathrm{2}^{\mathrm{n}} } +\mathrm{2}^{\mathrm{2}^{\mathrm{n}+\mathrm{1}} } \equiv\mathrm{1}+\left(−\mathrm{1}\right)^{\mathrm{2}^{\mathrm{n}} } +\left(−\mathrm{1}\right)^{\mathrm{2}^{\mathrm{n}+\mathrm{1}} } \equiv\mathrm{1}+\mathrm{1}+\mathrm{1}\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{3}\right)…
Question Number 126952 by Study last updated on 25/Dec/20 $$\underset{{n}=\mathrm{1}} {\overset{\mathrm{10000}} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}=??? \\ $$ Commented by Study last updated on 25/Dec/20 $${help}\:{me} \\ $$ Answered…
Question Number 192481 by Tomal last updated on 19/May/23 Answered by aleks041103 last updated on 19/May/23 $$\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{1}+{x}}\left(\mathrm{1}+\left(\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{1}+{x}}\right)^{\mathrm{4}} +…\right) \\ $$$${if}\:\:\mid\frac{\mathrm{1}}{\mathrm{1}+{x}}\mid<\mathrm{1},\:{i}.{e}.\:\mid\mathrm{1}+{x}\mid>\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{1}+{x}}\:\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }}=\frac{\mathrm{1}+{x}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} −\mathrm{1}}…
Question Number 192477 by Shrinava last updated on 19/May/23 $$\mathrm{Find}:\:\:\:\:\:\frac{\frac{\mathrm{25}}{\mathrm{42}}\:−\:\frac{\mathrm{5}}{\mathrm{16}}\:+\:\frac{\mathrm{10}}{\mathrm{9}}\:−\:\frac{\mathrm{2}}{\mathrm{3}}}{\frac{\mathrm{3}}{\mathrm{8}}\:+\:\frac{\mathrm{4}}{\mathrm{5}}\:−\:\frac{\mathrm{5}}{\mathrm{7}}\:−\:\frac{\mathrm{4}}{\mathrm{3}}}\:=\:? \\ $$ Answered by Tomal last updated on 19/May/23 $$=\frac{\left(\frac{\mathrm{25}}{\mathrm{42}}−\frac{\mathrm{5}}{\mathrm{16}}\right)+\left(\frac{\mathrm{10}}{\mathrm{9}}−\frac{\mathrm{2}}{\mathrm{3}}\right)}{\left(\frac{\mathrm{315}+\mathrm{672}−\mathrm{600}−\mathrm{1120}}{\mathrm{840}}\right)} \\ $$$$=\frac{\left(\frac{\mathrm{200}−\mathrm{105}}{\mathrm{336}}\right)+\left(\frac{\mathrm{10}−\mathrm{6}}{\mathrm{9}}\right)}{−\frac{\mathrm{733}}{\mathrm{840}}} \\ $$$$=\frac{\frac{\mathrm{95}}{\mathrm{336}}+\frac{\mathrm{4}}{\mathrm{9}}}{−\frac{\mathrm{733}}{\mathrm{840}}} \\…
Question Number 126935 by Raxreedoroid last updated on 25/Dec/20 $${Prove}\:{or}\:{give}\:{a}\:{counter}\:{example}: \\ $$$$\left({a}+\mathrm{1}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\underset{{k}−\mathrm{1}} {\overset{{n}−\mathrm{1}} {\:}}\right){a}^{{k}−\mathrm{1}} \\ $$$$\left(\underset{{r}} {\overset{{n}} {\:}}\right)=\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$ Terms of…