https://file.aiquickdraw.com/mathbot/file/1726172154150dg3svv2r.jpg

Let's go through each of the equations one by one to solve for \( x \).

### a) 
\[
\left( \frac{2}{3} \right)^{12} = \left( \frac{3}{2} \right)^{-3x}
\]
To solve this, note that:
\[
\left( \frac{3}{2} \right) = \left( \frac{2}{3} \right)^{-1}
\]
Thus, the right-hand side can be rewritten as:
\[
\left( \frac{2}{3} \right)^{3x}
\]
Now, the equation becomes:
\[
\left( \frac{2}{3} \right)^{12} = \left( \frac{2}{3} \right)^{3x}
\]
Since the bases are the same, equate the exponents:
\[
12 = 3x
\]
Solving for \( x \):
\[
x = \frac{12}{3} = 4
\]

---

### b) 
\[
((-8)^2)^3 = (-8)^{2x}
\]
Simplify the left-hand side:
\[
((-8)^2)^3 = (-8)^{6}
\]
Now the equation becomes:
\[
(-8)^6 = (-8)^{2x}
\]
Since the bases are the same, equate the exponents:
\[
6 = 2x
\]
Solving for \( x \):
\[
x = \frac{6}{2} = 3
\]

---

### c)
\[
\left( \frac{2}{3} \right)^3 \times \left( \frac{2}{3} \right)^{-6} = \left( \frac{2}{3} \right)^{x-4}
\]
Apply the law of exponents to the left-hand side:
\[
\left( \frac{2}{3} \right)^{3 + (-6)} = \left( \frac{2}{3} \right)^{-3}
\]
Now the equation becomes:
\[
\left( \frac{2}{3} \right)^{-3} = \left( \frac{2}{3} \right)^{x-4}
\]
Since the bases are the same, equate the exponents:
\[
-3 = x - 4
\]
Solving for \( x \):
\[
x = -3 + 4 = 1
\]

---

### d)
\[
\left( \left( \frac{3}{4} \right)^{-6} \div \left( \frac{3}{4} \right)^3 \right)^x = \left( \frac{3}{4} \right)^{-9}
\]
First, simplify the division inside the parentheses:
\[
\left( \frac{3}{4} \right)^{-6} \div \left( \frac{3}{4} \right)^3 = \left( \frac{3}{4} \right)^{-6 - 3} = \left( \frac{3}{4} \right)^{-9}
\]
Now the equation becomes:
\[
\left( \left( \frac{3}{4} \right)^{-9} \right)^x = \left( \frac{3}{4} \right)^{-9}
\]
Since the bases are the same, equate the exponents:
\[
-9x = -9
\]
Solving for \( x \):
\[
x = 1
\]

---

### Summary of solutions:
- a) \( x = 4 \)
- b) \( x = 3 \)
- c) \( x = 1 \)
- d) \( x = 1 \)

Would you like further clarification on any step or additional details?

---

Here are five related questions to deepen your understanding:
1. How would you solve a similar problem if the bases were not the same initially?
2. What exponent rules are crucial when solving equations like these?
3. How can you check your solutions to these types of equations?
4. How does changing the base of an exponential equation affect the solution?
5. Can you generalize a method for solving equations where the exponents are variables?

**Tip:** Always remember to simplify the bases and use exponent rules such as product and quotient of powers when solving exponential equations.

Find the value of x in the following exponential equations: (2/3)^12 = (3/2)^-3x, ((-8)^2)^3 = (-8)^2x, (2/3)^3 * (2/3)^-6 = (2/3)^(x-4), ((3/4)^-6 ÷ (3/4)^3)^x = (3/4)^-9

Step-by-step solutions to exponential equations with rational bases. This problem covers key algebraic concepts like exponent rules, product of powers, and solving for variables in exponentiation. Suitable for Grades 9-11.

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