Example 30. Solve the equation

$$
\sqrt[3]{(x+1)^{2}}+2 \sqrt[3]{(x-1)^{2}}=3 \sqrt[3]{x^{2}-1}
$$
Solution. The domain of definition (DOD) of equation (18) is the set of all real numbers. Note that if we let $u=\sqrt[3]{x+1}, v=$ $=\sqrt[3]{x-1}$, then equation (18) will take the form

$$
u^{2}+2 v^{2}-3 u v=0
$$

The left side of this equation is a homogeneous function of degree 2.

The number $x=1$ is not a root of equation (18); therefore, if $x \neq 1$, then $v \neq 0$. Dividing (19) by $v^{2}$ and setting $w=u / v$, we get the equation $w^{2}-3 w+2=0$, from which $w_{1}=1$ and $w_{2}=2$.

Thus, equation (18) on the remaining part of its DOD (i.e., for $x \neq-1$) is equivalent to the system

$$
\left[\begin{array}{l}
\frac{x+1}{x-1}=1 \\
\frac{x+1}{x-1}=8
\end{array}\right.
$$

The first equation of this system has no roots, the second has a unique solution $x=9 / 7$, which belongs to the DOD of equation (18). Therefore, equation (18) has a unique root $x=9 / 7$.