Solve in $\mathbb R$ the following equation. $$\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$$
Solution
Setting $x=2\times 10^{\frac 1{14}}y$, the equation becomes $64y^7-112y^5+56y^3-7y=\frac{7}{2\sqrt{10}}$
If $|y|\le 1$, then $y=\cos u$ for some $u$ and equation is $\cos 7u=\frac{7}{2\sqrt{10}}>1$, impossible.
If $y<-1$, then $y=-\cosh u$ for some $u$ and equation is $\cosh 7u=-\frac{7}{2\sqrt{10}}<0$, impossible.
If $y>1$, then $y=\cosh u$ for some $u$ and equation is $\cosh 7u=\frac{7}{2\sqrt{10}}$ and so $y=\cosh(\frac{\cosh^{-1} \frac{7}{2\sqrt{10}}}7)$
It's then easy to get $\cosh^{-1} \frac{7}{2\sqrt{10}}=\ln\frac{\sqrt{10}}2$ and so $y=2^{-\frac 87}10^{\frac 1{14}}+2^{-\frac 67}10^{-\frac 1{14}}$
Hence the unique root $\boxed{x=\sqrt[7]5+\sqrt[7]2}$
My question: Could you please solve by using simpler method?