Răspuns:
Explicație pas cu pas:
[tex]\texttt{Ridicam ecuatia la puterea a 3-a:}\\(\sqrt[3]{1+\sqrt x}+\sqrt[3]{8-\sqrt x})^3=3^3\\1+\sqrt x+8-\sqrt x+3\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}\underbrace{(\sqrt[3]{1+\sqrt x}+\sqrt[3]{8-\sqrt x})}=27\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=3\\ 9+3\cdot 3\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=27\\9+9\cdot \sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=27|:9\\1+\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=3\\\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=2|()^3\\(1+\sqrt{x})(8-\sqrt x)=8\\8-\sqrt{x}+8\sqrt{x}-x=8[/tex]
[tex]7\sqrt{x}-x=0\\ \sqrt{x}\stackrel{\texttt{not}}{=} t,t\geqslant 0\\7t-t^2=0\\t(7-t)=0\\t=0\Rightarrow x_1=0\\t=7\Rightarrow x_2=49\\\boxed{x_1+x_2=49}[/tex]
