$6.2955 \sqrt[3]{x \sqrt[5]{x}}+3 \sqrt[5]{x \sqrt[3]{x}}=8$
Solution. $5 x^{\frac{1}{3}} \cdot x^{\frac{1}{15}}+3 x^{\frac{1}{5}} \cdot x^{\frac{1}{15}}=8 \Leftrightarrow 5 x^{\frac{6}{15}}+3 x^{\frac{4}{15}}-8=0$.

Let $y=x^{\frac{2}{15}}$, then $5 y^{3}+3 y^{2}-8=0 \Leftrightarrow\left(5 y^{3}-5\right)+\left(3 y^{2}-3\right)=0 \Leftrightarrow$ $\Leftrightarrow(y-1)\left(5 y^{2}+8 y+8\right)=0 \Leftrightarrow y=1 \Leftrightarrow \sqrt[15]{x^{2}}=1 \Leftrightarrow x= \pm 1$.

Answer: $x_{1}=1, x_{2}=-1$.