Question

Rezolvati ecuatia:
$\sqrt[]{x} + \sqrt[5]{x} = \sqrt[9]{x+511}$

Answer 1

$\displaystyle Conditia~de~existenta~este~x \ge 0. \\ \\ Notam~x=y^{10},~y \ge 0,~iar~ecuatia~devine~y^5+y^2= \sqrt[9]{y^{10}+511} . \\ \\ Observam~ca~y=0~nu~convine. \\ \\ Impartind~prin~y^2,~obtinem~ y^3+1= \sqrt[9]{ \frac{1}{y^8}+511 \cdot \frac{1}{y^{18}}} . \\ \\ Consideram~functiile~f,g:(0,+\infty) \rightarrow \mathbb{R},~f(t)=t^3+1, \\ \\ g(t)=\sqrt[9]{ \frac{1}{t^8}+511 \cdot \frac{1}{t^{18}}}. \\ \\ Evident~f~este~strict~crescatoare,~iar~g~este~strict~descrescatoare.$

$\displaystyle Prin~urmare,~ecuatia~f(y)=g(y)~admite~cel~mult~o~solutie. \\ \\ Observam~ca~y=1~este~solutie,~ceea~ce~inseamna~ca~aceasta~este \\ \\ unica~solutie. \\ \\ x=y^{10}=1. \\ \\ Solutia~este,~deci,~x=1.$