6.060. $\sqrt[3]{5 x+7}-\sqrt[3]{5 x-12}=1$.

Solve the equation $\sqrt[3]{5 x+7}-\sqrt[3]{5 x-12}=1$.
## Solution.

Let's rewrite the equation as $\sqrt[3]{5 x+7}=\sqrt[3]{5 x-12}+1$ and cube both sides:

$$
\begin{aligned}
& 5 x+7=5 x-12+3(\sqrt[3]{5 x-12})^{2}+3 \sqrt[3]{5 x-12}+1 \Leftrightarrow \\
& \Leftrightarrow(\sqrt[3]{5 x-12})^{2}+\sqrt[3]{5 x-12}-6=0
\end{aligned}
$$

Let $\sqrt[3]{5 x-12}=t$. The equation in terms of $t$ becomes $t^{2}-t-6=0$, from which we find $t_{1}=-3$ and $t_{2}=2$.

Then either $\sqrt[3]{5 x-12}=-3, 5 x-12=-27, x_{1}=-3$, or $\sqrt[3]{5 x-12}=2$, $5 x-12=8, x_{2}=4$.

Answer: $x_{1}=-3, x_{2}=4$.