Rewrite the equation with a common base.

Step 1: Rewrite the equation with a common base.
The given equation is:

$$\left(\frac{27}{8}\right)^{x+7} - \left(\frac{3}{2}\right)^{-3x} = 0$$

Rearrange the equation:

$$\left(\frac{27}{8}\right)^{x+7} = \left(\frac{3}{2}\right)^{-3x}$$

Notice that $\frac{27}{8}$ can be written as $\left(\frac{3}{2}\right)^3$:

$$\left(\left(\frac{3}{2}\right)^3\right)^{x+7} = \left(\frac{3}{2}\right)^{-3x}$$

Step