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Question-204349

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Question Number 204349 by Abdullahrussell last updated on 14/Feb/24
Commented by mr W last updated on 14/Feb/24
you can′t prove something wrong!
$${you}\:{can}'{t}\:{prove}\:{something}\:{wrong}! \\ $$
Commented by mr W last updated on 14/Feb/24
m=(((√6)+(√5))/2)<(((√9)+(√9))/2)=3 ⇒m^5 <3^5 =243 m=(((√6)+(√5))/2)>(((√4)+(√4))/2)=2 m^5 >2^5 =32 ⇒((32)/m^5 )<1 ⇒m^5 +((32)/m^5 )<243+1=244 but 1048(√6)>1048(√4)=2096 ⇒m^5 +((32)/m^5 ) can never be =1048(√6) !
$${m}=\frac{\sqrt{\mathrm{6}}+\sqrt{\mathrm{5}}}{\mathrm{2}}<\frac{\sqrt{\mathrm{9}}+\sqrt{\mathrm{9}}}{\mathrm{2}}=\mathrm{3} \\ $$$$\Rightarrow{m}^{\mathrm{5}} <\mathrm{3}^{\mathrm{5}} =\mathrm{243} \\ $$$${m}=\frac{\sqrt{\mathrm{6}}+\sqrt{\mathrm{5}}}{\mathrm{2}}>\frac{\sqrt{\mathrm{4}}+\sqrt{\mathrm{4}}}{\mathrm{2}}=\mathrm{2} \\ $$$${m}^{\mathrm{5}} >\mathrm{2}^{\mathrm{5}} =\mathrm{32} \\ $$$$\Rightarrow\frac{\mathrm{32}}{{m}^{\mathrm{5}} }<\mathrm{1} \\ $$$$\Rightarrow{m}^{\mathrm{5}} +\frac{\mathrm{32}}{{m}^{\mathrm{5}} }<\mathrm{243}+\mathrm{1}=\mathrm{244} \\ $$$${but}\:\mathrm{1048}\sqrt{\mathrm{6}}>\mathrm{1048}\sqrt{\mathrm{4}}=\mathrm{2096} \\ $$$$\Rightarrow{m}^{\mathrm{5}} +\frac{\mathrm{32}}{{m}^{\mathrm{5}} }\:\:{can}\:{never}\:{be}\:=\mathrm{1048}\sqrt{\mathrm{6}}\:! \\ $$
Answered by Rasheed.Sindhi last updated on 14/Feb/24
m+n=(√6) & m−n=(√5) ⇒m=(((√6) +(√5) )/2) ⇒(2/m)=(2/(((√6) +(√5) )/2))=(4/( (√6) +(√5) ))∙(((√6) −(√5))/( (√6) −(√5))) =((4((√6) −(√5) ))/1)=4((√6) −(√5) ) ⇒m+(2/m)=(((√6) +(√5) )/2)+4((√6) −(√5) ) =((5(√6) −3(√5) )/2) ⇒(m+(2/m))^2 =(((5(√6) −3(√5) )/2))^2 m^2 +(4/m^2 )=(((5(√6) −3(√5) )/2))^2 −4=((25(6)+9(5)−30(√(30)) −16)/4)=((179−30(√(30)))/4) (m+(2/m))^3 =(((5(√6) −3(√5) )/2))^3 m^3 +(8/m^3 )=((1425(√6) −1485(√5))/8)−6(((5(√6) −3(√5) )/2)) =((1425(√6) −1485(√5)−24(5(√6) −3(√5) ))/8) =((1425(√6) −1485(√5)−120(√6) +72(√5) ))/8) =((1305(√6) −13(√5))/8) (m^2 +(4/m^2 ))(m^3 +(8/m^3 ))=(((179−30(√(30)))/4))(((1305(√6) −13(√5))/8)) m^5 +((32)/m^3 )+4(m+(2/m))=(((179−30(√(30)))(1305(√6) −13(√5)))/(32)) m^5 +((32)/m^3 )+4(((5(√6) −3(√5) )/2))=(((179−30(√(30)))(1305(√6) −13(√5)))/(32)) m^5 +((32)/m^3 )=(((179−30(√(30)))(1305(√6) −13(√5)))/(32))−4(((5(√6) −3(√5) )/2)) Continue
$${m}+{n}=\sqrt{\mathrm{6}}\:\:\:\:\&\:\:\:{m}−{n}=\sqrt{\mathrm{5}}\: \\ $$$$\Rightarrow{m}=\frac{\sqrt{\mathrm{6}}\:\:+\sqrt{\mathrm{5}}\:}{\mathrm{2}} \\ $$$$\Rightarrow\frac{\mathrm{2}}{{m}}=\frac{\mathrm{2}}{\frac{\sqrt{\mathrm{6}}\:+\sqrt{\mathrm{5}}\:}{\mathrm{2}}}=\frac{\mathrm{4}}{\:\sqrt{\mathrm{6}}\:+\sqrt{\mathrm{5}}\:}\centerdot\frac{\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{5}}}{\:\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{5}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{4}\left(\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{5}}\:\right)}{\mathrm{1}}=\mathrm{4}\left(\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{5}}\:\right) \\ $$$$\Rightarrow{m}+\frac{\mathrm{2}}{{m}}=\frac{\sqrt{\mathrm{6}}\:\:+\sqrt{\mathrm{5}}\:}{\mathrm{2}}+\mathrm{4}\left(\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{5}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}} \\ $$$$\Rightarrow\left({m}+\frac{\mathrm{2}}{{m}}\right)^{\mathrm{2}} =\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${m}^{\mathrm{2}} +\frac{\mathrm{4}}{{m}^{\mathrm{2}} }=\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{4}=\frac{\mathrm{25}\left(\mathrm{6}\right)+\mathrm{9}\left(\mathrm{5}\right)−\mathrm{30}\sqrt{\mathrm{30}}\:−\mathrm{16}}{\mathrm{4}}=\frac{\mathrm{179}−\mathrm{30}\sqrt{\mathrm{30}}}{\mathrm{4}}\: \\ $$$$\left({m}+\frac{\mathrm{2}}{{m}}\right)^{\mathrm{3}} =\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$${m}^{\mathrm{3}} +\frac{\mathrm{8}}{{m}^{\mathrm{3}} }=\frac{\mathrm{1425}\sqrt{\mathrm{6}}\:−\mathrm{1485}\sqrt{\mathrm{5}}}{\mathrm{8}}−\mathrm{6}\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1425}\sqrt{\mathrm{6}}\:−\mathrm{1485}\sqrt{\mathrm{5}}−\mathrm{24}\left(\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\left.\mathrm{1425}\sqrt{\mathrm{6}}\:−\mathrm{1485}\sqrt{\mathrm{5}}−\mathrm{120}\sqrt{\mathrm{6}}\:+\mathrm{72}\sqrt{\mathrm{5}}\:\:\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1305}\sqrt{\mathrm{6}}\:−\mathrm{13}\sqrt{\mathrm{5}}}{\mathrm{8}} \\ $$$$\left({m}^{\mathrm{2}} +\frac{\mathrm{4}}{{m}^{\mathrm{2}} }\right)\left({m}^{\mathrm{3}} +\frac{\mathrm{8}}{{m}^{\mathrm{3}} }\right)=\left(\frac{\mathrm{179}−\mathrm{30}\sqrt{\mathrm{30}}}{\mathrm{4}}\right)\left(\frac{\mathrm{1305}\sqrt{\mathrm{6}}\:−\mathrm{13}\sqrt{\mathrm{5}}}{\mathrm{8}}\right) \\ $$$${m}^{\mathrm{5}} +\frac{\mathrm{32}}{{m}^{\mathrm{3}} }+\mathrm{4}\left({m}+\frac{\mathrm{2}}{{m}}\right)=\frac{\left(\mathrm{179}−\mathrm{30}\sqrt{\mathrm{30}}\right)\left(\mathrm{1305}\sqrt{\mathrm{6}}\:−\mathrm{13}\sqrt{\mathrm{5}}\right)}{\mathrm{32}} \\ $$$${m}^{\mathrm{5}} +\frac{\mathrm{32}}{{m}^{\mathrm{3}} }+\mathrm{4}\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right)=\frac{\left(\mathrm{179}−\mathrm{30}\sqrt{\mathrm{30}}\right)\left(\mathrm{1305}\sqrt{\mathrm{6}}\:−\mathrm{13}\sqrt{\mathrm{5}}\right)}{\mathrm{32}} \\ $$$${m}^{\mathrm{5}} +\frac{\mathrm{32}}{{m}^{\mathrm{3}} }=\frac{\left(\mathrm{179}−\mathrm{30}\sqrt{\mathrm{30}}\right)\left(\mathrm{1305}\sqrt{\mathrm{6}}\:−\mathrm{13}\sqrt{\mathrm{5}}\right)}{\mathrm{32}}−\mathrm{4}\left(\frac{\mathrm{5}\sqrt{\mathrm{6}}\:−\mathrm{3}\sqrt{\mathrm{5}}\:\:}{\mathrm{2}}\right) \\ $$$${Continue} \\ $$

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