Example 22. Solve the equation

$$
3^{x} \cdot 8^{x /(x+1)}=36
$$
Solution. We have:

$$
\begin{aligned}
3^{x} \cdot 2^{3 x /(x+1)}=3^{2} \cdot 2^{2} \Leftrightarrow & 3^{x-2} \cdot 2^{3 x /(x+1)-2}=1 \Leftrightarrow \\
& \Leftrightarrow 3^{x-2} \cdot 2^{(x-2) /(x+1)}=1 \Leftrightarrow\left(3 \cdot 2^{1 /(x+1)}\right)^{x-2}=1
\end{aligned}
$$

Thus, equation (24) is equivalent to the equation

$$
10^{(x-2) \lg \left(3 \cdot 2^{1 /(1+x)}\right)}=1 \text {; }
$$

from which

$$
(x-2) \lg \left(3 \cdot 2^{1 /(x+1)}\right)=0 \text {. }
$$

Therefore, the original equation is equivalent to the system of equations

$$
x-2=0, \quad \lg \left(3 \cdot 2^{1 /(x+1)}\right)=0
$$

solving which we find two roots of equation (24): $x_{1}=2, x_{2}=$ $=-1-\lg _{3} 2$.