6.370 Solve the equation $\sqrt[4]{x^{4}+x-2}+\sqrt{x^{4}+x-2}=6$ given that $\boldsymbol{x}>\mathbf{0}$.
Solution. Let $y=\sqrt[4]{x^{4}+x-2}$, then $y^{2}+y-6=0 \Leftrightarrow \sqrt[4]{x^{4}+x-2}=2 \Leftrightarrow$ $\Leftrightarrow x^{4}+x-18=0 \Leftrightarrow(x-2)\left(x^{3}+2 x^{2}+4 x+9\right)=0$.

Since $x>0$, then $x^{3}+2 x^{2}+4 x+9>0 \Rightarrow x=2$.

Answer: $x=2$.

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