6.159. $\sqrt{x+2}-\sqrt[3]{3 x+2}=0$.

6.159. $\sqrt{x+2}-\sqrt[3]{3 x+2}=0$.
## Solution.

Domain of definition: $x+2 \geq 0, x \geq-2$.

$$
\begin{aligned}
& \text { From the condition } \sqrt{x+2}=\sqrt[3]{3 x+2} \Rightarrow(x+2)^{3}=(3 x+2)^{2} \\
& x^{3}+6 x^{2}+12 x+8=9 x^{2}+12 x+4 \Leftrightarrow x^{3}-3 x^{2}+4=0 \Leftrightarrow \\
& \Leftrightarrow(x+1)\left(x^{2}-x+1\right)-3(x-1)(x+1)=0 \Leftrightarrow(x+1)\left(x^{2}-4 x+4\right)=0 \\
& (x+1)(x-2)^{2}=0 . \text { Hence } x+1=0, x_{1}=-1, \text { or }(x-2)^{2}=0, x_{2,3}=2
\end{aligned}
$$

By substituting into the original equation, we verify that $x_{1}=-1$ is an extraneous root.

Answer: $x=2$.