This equation involves exponent rules, and the base for each term is $\frac{-8}{5}$. Here's how to approach the solution step-by-step:

Given Equation:

$\left( \frac{-8}{5} \right)^4 \cdot \left( \frac{-8}{5} \right)^{x+5} = \left( \frac{-8}{5} \right)^{2x+3}$

Step 1: Combine the Exponents on the Left

Using the rule of exponents $a^m \cdot a^n = a^{m+n}$, we combine the exponents of the terms on the left-hand side: $\left( \frac{-8}{5} \right)^{4 + (x+5)} = \left( \frac{-8}{5} \right)^{2x+3}$

Simplify the exponents: $4 + x + 5 = x + 9$

Thus, the left-hand side becomes: $\left( \frac{-8}{5} \right)^{x+9}$

Step 2: Equate the Exponents

Since the bases are the same, the exponents must also be equal: $x + 9 = 2x + 3$

Step 3: Solve for $x$

Simplify the equation: $9 - 3 = 2x - x$

$6 = x$

Final Solution:

$x = 6$

Would you like a more detailed explanation or have questions about a specific step?

Here are 5 related questions for further practice:

1. How would the solution change if the base were different?
2. What are the rules for dividing exponents with the same base?
3. How do you solve equations with fractional exponents?
4. What happens if $x$ is negative in an exponential equation?
5. Can exponential equations have no solution? Why?

Tip: Always combine like terms and simplify step-by-step to avoid errors in exponential equations!