$6.290 \sqrt{x+2 \sqrt{x-1}}+\sqrt{x-2 \sqrt{x-1}}=x-1$.
Solution. $\sqrt{(x-1)+2 \sqrt{x-1}+1}+\sqrt{(x-1)-2 \sqrt{x-1}+1}=x-1 \Leftrightarrow$ $\Leftrightarrow \sqrt{(\sqrt{x-1}+1)^{2}}+\sqrt{(\sqrt{x-1}-1)^{2}}=x-1 \Leftrightarrow \sqrt{x-1}+1+|\sqrt{x-1}-1|=x-1 \Leftrightarrow$ $\Leftrightarrow \sqrt{x-1}+|\sqrt{x-1}-1|=x-2$.

Since $x \geq 2$, then $\sqrt{x-1} \geq 1$. Therefore, the last equation is equivalent to the system:

$\left\{\begin{array}{l}2 \sqrt{x-1}-1=x-2, \\ x \geq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}2 \sqrt{x-1}=x-1, \\ x \geq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}2=\sqrt{x-1}, \\ x \geq 2\end{array} \Leftrightarrow x=5\right.\right.\right.$.

Answer: $x=5$.

$6.2918,4 \sqrt[12]{x^{-7}}-0.2 \sqrt[4]{x^{-1} \cdot \sqrt[3]{x^{2}}}=\sqrt[12]{x^{11}}$

Solution. Note that $\sqrt[4]{x^{-1} \cdot \sqrt[3]{x^{2}}}=\sqrt[12]{x^{-1}}$. Multiply both sides of the equation by $\sqrt[12]{x}: \frac{8.4}{\sqrt{x}}-0.2=x$. Let $y=\sqrt{x}$, then $5 y^{3}+y-42=0 \Leftrightarrow$ $\Leftrightarrow(y-2)\left(5 y^{2}+10 y+21\right)=0 \Leftrightarrow y=2 \Leftrightarrow x=4$.

Answer: $x=4$.