Question Number 169989 by udaythool last updated on 13/May/22 $$ \\ $$$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}: \\ $$$$\mathrm{For}\:\mathrm{any}\:\mathrm{extension}\:{E}\:\mathrm{of}\:\mathrm{a}\:\mathrm{field}\:{F}, \\ $$$${F}\left({u}\right)={F}\left[{u}\right]\:\:\:\:\forall\:{u}\in{E}. \\ $$$$\mathrm{Where}\:{F}\left({u}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{smallest}\:\mathrm{subfield} \\ $$$$\mathrm{of}\:{E}\:\mathrm{containing}\:{F}\:\mathrm{and}\:{u}\:\mathrm{and} \\ $$$${F}\left[{u}\right]=\left\{{f}\left({u}\right)\mid{f}\left({x}\right)\in{F}\left[{x}\right]\right\},\:{F}\left[{x}\right]\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{polynomial}\:\mathrm{ring}\:\mathrm{over}\:{F}. \\…
Question Number 104450 by Anindita last updated on 21/Jul/20 $$\frac{\mathrm{4}}{\mathrm{99}}\:+\:\frac{\mathrm{7}}{\mathrm{999}}\:+\:\frac{\mathrm{11}}{\mathrm{999999}}\:=\:? \\ $$$$\boldsymbol{\mathrm{Can}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{this}}? \\ $$ Answered by OlafThorendsen last updated on 21/Jul/20 $$\mathrm{999999}\:=\:\mathrm{999000}+\mathrm{999}\:=\:\mathrm{999}×\mathrm{1001} \\ $$$$\mathrm{999999}\:=\:\mathrm{990000}+\mathrm{9900}+\mathrm{99}\:=\:\mathrm{99}×\mathrm{10101} \\…
Question Number 38907 by ajfour last updated on 01/Jul/18 $$\sqrt{{c}^{\mathrm{2}} +{x}^{\mathrm{2}} }\:=\:\mathrm{1}+\frac{{c}}{{x}} \\ $$$${how}\:{many}\:{complex}\:{roots}\:? \\ $$ Commented by ajfour last updated on 01/Jul/18 Commented by…
Question Number 169977 by infinityaction last updated on 13/May/22 Answered by ajfour last updated on 13/May/22 $$\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} \geqslant\mathrm{16}\sqrt{{abcd}} \\ $$$$\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)\geqslant\mathrm{4}\sqrt{{abcd}} \\…
Question Number 104419 by Anindita last updated on 21/Jul/20 $$\boldsymbol{\mathrm{Now}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{is}} \\ $$$$\mathrm{90}\:\boldsymbol{\mathrm{years}}.\:\mathrm{5}\:\boldsymbol{\mathrm{years}}\:\boldsymbol{\mathrm{ago}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{age}} \\ $$$$\boldsymbol{\mathrm{was}}\:\mathrm{3}\:\boldsymbol{\mathrm{times}}\:\boldsymbol{\mathrm{than}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{age}}.\:\boldsymbol{\mathrm{Write}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{age}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{father}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{age}}\:\mathrm{3}\:\boldsymbol{\mathrm{years}} \\ $$$$\boldsymbol{\mathrm{after}}. \\ $$ Answered by 1549442205PVT last updated…
Question Number 169949 by Shrinava last updated on 13/May/22 Commented by mr W last updated on 13/May/22 $${if}\:{F}\:{is}\:{area},\:{then}\:{question}\:{must}\:{be} \\ $$$${wrong}. \\ $$$${LHS}\:{is}\:\left[{length}\right]^{\mathrm{4}} \\ $$$${RHS}\:{is}\:\left[{length}\right]^{\mathrm{6}} \:!…
Question Number 104413 by Anindita last updated on 21/Jul/20 $$\boldsymbol{\mathrm{The}}\:\boldsymbol{\mathrm{sum}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{two}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{is}}\:\mathrm{304} \\ $$$$\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{GCD}}\:\boldsymbol{\mathrm{is}}\:\mathrm{19}.\:\boldsymbol{\mathrm{What}}\:\boldsymbol{\mathrm{are}}\:\boldsymbol{\mathrm{the}} \\ $$$$\boldsymbol{\mathrm{numbers}}? \\ $$ Answered by Rasheed.Sindhi last updated on 22/Jul/20 $${Integers}:\:\mathrm{19}{x},\mathrm{19}{y} \\…
Question Number 169945 by mathlove last updated on 13/May/22 Commented by mathlove last updated on 13/May/22 $${help}\:{me} \\ $$ Commented by Rasheed.Sindhi last updated on…
Question Number 104410 by Anindita last updated on 21/Jul/20 $$\boldsymbol{\mathrm{If}}\:\mathrm{11}×\mathrm{11}=\mathrm{121}, \\ $$$$\mathrm{111}×\mathrm{111}=\mathrm{12321},\mathrm{1111}×\mathrm{1111}=\mathrm{1234321} \\ $$$$\boldsymbol{\mathrm{then}}\:\boldsymbol{\mathrm{what}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{answer}}\:\boldsymbol{\mathrm{of}} \\ $$$$\mathrm{111111}×\mathrm{111111}\:? \\ $$$$\left(\boldsymbol{\mathrm{It}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{short}}\:\boldsymbol{\mathrm{question}}\right) \\ $$ Answered by OlafThorendsen last updated…
Question Number 104404 by Anindita last updated on 21/Jul/20 $$\frac{\mathrm{4}}{\mathrm{5}}=\frac{?}{\mathrm{45}}=\frac{\mathrm{20}}{?}=\frac{\mathrm{144}}{?}=\frac{\mathrm{200}}{?} \\ $$ Answered by Don08q last updated on 21/Jul/20 $$\frac{\mathrm{4}}{\mathrm{5}}=\frac{\mathrm{9}×\mathrm{4}}{\mathrm{45}}=\frac{\mathrm{20}}{\mathrm{5}×\mathrm{5}}=\frac{\mathrm{144}}{\mathrm{5}×\mathrm{36}}=\frac{\mathrm{200}}{\mathrm{5}×\mathrm{50}}\:_{} \\ $$$$\frac{\mathrm{4}}{\mathrm{5}}=\frac{\mathrm{36}}{\mathrm{45}}=\frac{\mathrm{20}}{\mathrm{25}}=\frac{\mathrm{144}}{\mathrm{180}}=\frac{\mathrm{200}}{\mathrm{250}} \\ $$$$ \\…