We are tasked with solving the equation:

$(-2)^3 \times (-2)^6 = (-2)^{2x - 1}$

Step 1: Apply the properties of exponents

Recall the property of exponents: when multiplying powers with the same base, we add the exponents:

$a^m \times a^n = a^{m+n}$

In our case, the base is $-2$, so we can rewrite the left-hand side as:

$(-2)^3 \times (-2)^6 = (-2)^{3 + 6} = (-2)^9$

Thus, the equation becomes:

$(-2)^9 = (-2)^{2x - 1}$

Step 2: Set the exponents equal

Since the bases are the same, we can equate the exponents:

$9 = 2x - 1$

Step 3: Solve for $x$

Now, solve the equation $9 = 2x - 1$ for $x$:

$9 + 1 = 2x \quad \text{(add 1 to both sides)}$ $10 = 2x$ $x = \frac{10}{2} = 5$

Thus, the value of $x$ is $\boxed{5}$.

---

Would you like a more detailed explanation, or do you have any questions about this solution?

Here are 5 related questions:

1. What is the value of $(-2)^9$?
2. How do you handle negative bases when raising them to powers?
3. Can the same exponent rule apply to fractions or decimals with negative bases?
4. How do you solve exponential equations with different bases?
5. What is the general rule for multiplying powers with the same base?

Tip: Always remember that when you multiply powers with the same base, simply add the exponents.