Question Number 174288 by Engr_Jidda last updated on 28/Jul/22
Commented by Engr_Jidda last updated on 28/Jul/22
$${find}\:{the}\:{value}\:{of}\:{t} \\ $$
Commented by a.lgnaoui last updated on 30/Jul/22
$${the}\:{idea}\:{is}\:{to}\:{transforme} \\ $$$$\mathrm{65536}{t}^{\mathrm{3}} +{t}^{\mathrm{2}} −\mathrm{131076}\:\:\:{to} \\ $$$${the}\:{expression}\:{like} \\ $$$${Z}^{\mathrm{3}} −{pZ}+{q}=\mathrm{0} \\ $$$$\mathrm{256}^{\mathrm{2}} {t}^{\mathrm{3}} +{t}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{256}^{\mathrm{2}} +\mathrm{2}\right)=\mathrm{0} \\ $$$${a}=\mathrm{256}^{\mathrm{2}} \\ $$$${t}^{\mathrm{3}} +\frac{{t}^{\mathrm{2}} }{{a}^{\mathrm{2}} }−\mathrm{2}\left({a}+\mathrm{2}\right)=\mathrm{0} \\ $$$${t}^{\mathrm{3}} +\mathrm{3}\left(\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right){t}^{\mathrm{2}} −\mathrm{2}\left({a}+\mathrm{2}\right)=\mathrm{0} \\ $$$$\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)^{\mathrm{3}} −\mathrm{3}\left(\frac{\mathrm{1}}{\mathrm{9}{a}^{\mathrm{4}} }\right){t}−\frac{\mathrm{1}}{\mathrm{27}{a}^{\mathrm{6}} }−\mathrm{2}\left({a}+\mathrm{2}\right)=\mathrm{0} \\ $$$$\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{4}} }\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)+\frac{\mathrm{1}}{\mathrm{9}{a}^{\mathrm{6}} }−\frac{\mathrm{1}}{\mathrm{27}{a}^{\mathrm{6}} }−\mathrm{2}\left({a}+\mathrm{2}\right)=\mathrm{0} \\ $$$$\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{4}} }\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)+\frac{\mathrm{2}}{\mathrm{27}{a}^{\mathrm{6}} }−\mathrm{2}\left({a}+\mathrm{2}\right)=\mathrm{0} \\ $$$${Z}=\left({t}+\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\right)={t}+\frac{\mathrm{1}}{\mathrm{196608}}\:\:\:{with}\:{a}=\mathrm{256} \\ $$$${Z}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }{Z}+\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{27}{a}^{\mathrm{6}} }−{a}−\mathrm{2}\right)=\mathrm{0} \\ $$$${p}=\frac{\mathrm{1}}{\mathrm{3}{a}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:{q}=\mathrm{2}\left(\frac{\mathrm{1}}{\mathrm{27}{a}^{\mathrm{6}} }−{a}−\mathrm{2}\right) \\ $$$${the}\:\:{rest}\:{to}\:{continue}….. \\ $$$${the}\:{pricessus}\:{is}\:{long}! \\ $$$${excuse}\:{me}\:. \\ $$$$ \\ $$$$ \\ $$
Answered by mr W last updated on 30/Jul/22
$$\frac{\mathrm{1}}{{t}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{131076}{t}}−\frac{\mathrm{65536}}{\mathrm{131076}}=\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{t}}=\sqrt[{\mathrm{3}}]{\sqrt{\left(\frac{\mathrm{32768}}{\mathrm{131076}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{393228}^{\mathrm{3}} }}+\frac{\mathrm{32768}}{\mathrm{131076}}}−\sqrt[{\mathrm{3}}]{\sqrt{\left(\frac{\mathrm{32768}}{\mathrm{131076}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{393228}^{\mathrm{3}} }}−\frac{\mathrm{32768}}{\mathrm{131076}}} \\ $$$$\:\:\:\:\:\approx\mathrm{0}.\mathrm{793}\:\mathrm{695}\:\mathrm{756} \\ $$$$\Rightarrow{t}\approx\mathrm{1}.\mathrm{259}\:\mathrm{928}\:\mathrm{780} \\ $$
Commented by Tawa11 last updated on 31/Jul/22
$$\mathrm{Great}\:\mathrm{sir} \\ $$