Example 6. Solve the equation

$$
\sqrt[3]{2 x-1}+\sqrt[3]{x-1}=1
$$
Solution. Raising both sides of the equation to the third power, we get

$$
3 x-2+3 \sqrt[3]{(2 x-1)(x-1)}(\sqrt[3]{2 x-1}+\sqrt[3]{x-1})=1
$$

By the condition, the expression $\sqrt[3]{2 x-1}+\sqrt[3]{x-1}$ equals one. Substituting this expression with one in the obtained equation, we get the equation

$$
3 x-2+3 \sqrt[3]{(2 x-1)(x-1)}=1
$$

which is a consequence of the original equation, since equation (1) may have a root that does not necessarily satisfy the relation:

$$
1=\sqrt[3]{2 x-1}+\sqrt[3]{x-1}
$$

Raising equation (1) to the third power:

$$
(2 x-1)(x-1)=(1-x)^{3}
$$

The last equation has roots $x_{1}=0$ and $x_{2}=1$.

Verification shows that $x_{1}=0$ is an extraneous root of the original equation, while $x_{2}=1$ satisfies it.

Thus, $x_{2}=1$ is the only root of the original equation.

For an equation of the form

$$
\sqrt[3]{f(x)}+\sqrt[3]{g(x)}=\varphi(x)
$$

where $f(x), g(x), \varphi(x)$ are some functions, they are usually solved as follows. Both sides of the equation are raised to the third power to obtain the equation

$$
f(x)+g(x)+3 \sqrt[3]{f(x) g(x)}(\sqrt[3]{f(x)}+\sqrt[3]{g(x)})=\varphi^{3}(x)
$$

In this equation, the expression $\sqrt[3]{f(x)}+\sqrt[3]{g(x)}$, which is the left-hand side of the original equation, is replaced by $\varphi(x)$, which is its right-hand side. As a result, we get the equation

$$
f(x)+g(x)+3 \sqrt[3]{f(x) g(x)} \varphi(x)=\varphi^{3}(x)
$$

which (since it is solved under the condition $\sqrt[3]{f(x)}+\sqrt[3]{g(x)}=$ $=\varphi(x)$) is a consequence of the original equation. After isolating the radical and raising the obtained equation to the third power, we find its roots, and then by direct substitution of each of the found numbers into the original equation, we select those that are roots of the original equation.