Question-77819

Question Number 77819 by aliesam last updated on 10/Jan/20

Answered by lémùst last updated on 10/Jan/20

posons t=^3 (√r) , ainsi t − (1/t) = 1 ⇒t^2 −t−1=0 ⇒t=±((1+(√5))/2) donc r=±(((1+(√5))/2))^3 il vous reste juste a calculer r−(1/r) et r^3 −(1/r^3 ) en utilisans les regles de calculs usuelles

$${posons}\:{t}=\:^{\mathrm{3}} \sqrt{{r}}\:,\:{ainsi}\:{t}\:−\:\frac{\mathrm{1}}{{t}}\:=\:\mathrm{1} \\ $$$$\Rightarrow{t}^{\mathrm{2}} −{t}−\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{t}=\pm\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$${donc}\:{r}=\pm\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{\mathrm{3}} \\ $$$${il}\:{vous}\:{reste}\:{juste}\:{a}\:{calculer}\:{r}−\frac{\mathrm{1}}{{r}}\:{et}\:{r}^{\mathrm{3}} −\frac{\mathrm{1}}{{r}^{\mathrm{3}} } \\ $$$${en}\:{utilisans}\:{les}\:{regles}\:{de}\:{calculs}\:{usuelles} \\ $$

Answered by key of knowledge last updated on 11/Jan/20

r−(1/r)=(r^(1/3) −(1/r^(1/3) ))(r^(2/3) +(1/r^(2/3) )+1)= (r^(1/3) −(1/r^(1/3) ))((r^(1/3) −(1/r^(1/3) ))^2 +3)= (1)×(1^2 +3)=4 r^3 −(1/r^3 )=(r^1 −(1/r^1 ))((r^1 −(1/r^1 ))^2 +3)= (4)(4^2 +3)=76

$$\mathrm{r}−\frac{\mathrm{1}}{\mathrm{r}}=\left(\mathrm{r}^{\mathrm{1}/\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{1}/\mathrm{3}} }\right)\left(\mathrm{r}^{\mathrm{2}/\mathrm{3}} +\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{2}/\mathrm{3}} }+\mathrm{1}\right)= \\ $$$$\left(\mathrm{r}^{\mathrm{1}/\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{1}/\mathrm{3}} }\right)\left(\left(\mathrm{r}^{\mathrm{1}/\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{1}/\mathrm{3}} }\right)^{\mathrm{2}} +\mathrm{3}\right)= \\ $$$$\left(\mathrm{1}\right)×\left(\mathrm{1}^{\mathrm{2}} +\mathrm{3}\right)=\mathrm{4} \\ $$$$\mathrm{r}^{\mathrm{3}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{3}} }=\left(\mathrm{r}^{\mathrm{1}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{1}} }\right)\left(\left(\mathrm{r}^{\mathrm{1}} −\frac{\mathrm{1}}{\mathrm{r}^{\mathrm{1}} }\right)^{\mathrm{2}} +\mathrm{3}\right)= \\ $$$$\left(\mathrm{4}\right)\left(\mathrm{4}^{\mathrm{2}} +\mathrm{3}\right)=\mathrm{76} \\ $$

Commented by aliesam last updated on 10/Jan/20