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Question Number 194791 by dimentri last updated on 15/Jul/23
   x
$$\:\:\:\underline{\underbrace{\boldsymbol{{x}}}} \\ $$
Answered by Frix last updated on 15/Jul/23
If ((−r))^(1/7) =−(r)^(1/7)   ((x^2 −2)/(2x^2 ))((x−(√2)))^(1/7) =(x^(9/7) /(2((x+(√2)))^(1/7) ))  (x^2 −2)^(8/7) =x^((23)/7)   (x^2 −2)^8 =x^(23)   x=1
$$\mathrm{If}\:\sqrt[{\mathrm{7}}]{−{r}}=−\sqrt[{\mathrm{7}}]{{r}} \\ $$$$\frac{{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{2}{x}^{\mathrm{2}} }\sqrt[{\mathrm{7}}]{{x}−\sqrt{\mathrm{2}}}=\frac{{x}^{\frac{\mathrm{9}}{\mathrm{7}}} }{\mathrm{2}\sqrt[{\mathrm{7}}]{{x}+\sqrt{\mathrm{2}}}} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\frac{\mathrm{8}}{\mathrm{7}}} ={x}^{\frac{\mathrm{23}}{\mathrm{7}}} \\ $$$$\left({x}^{\mathrm{2}} −\mathrm{2}\right)^{\mathrm{8}} ={x}^{\mathrm{23}} \\ $$$${x}=\mathrm{1} \\ $$

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