Example 16. Solve the equation

$$
5 \cdot 2^{3 x-3}-3 \cdot 2^{5-3 x}+7=0
$$
Solution. Using the properties of exponents, rewrite the given equation as

$$
\frac{5}{8} \cdot 2^{3 x}-\frac{96}{2^{3 x}}+7=0
$$

This equation is of the form (12).

Let $t=2^{3 x}$; then we have $\frac{5}{8} t-\frac{96}{t}+7=0$, i.e., $5 t^{2}+56 t-768=0$. From this, we find $t_{1}=-96 / 5, t_{2}=8$.

Thus, equation (13) is equivalent to the equation

$$
2^{3 x}=8
$$

from which $x=1$.