The given equation is:

\[
\left( \frac{3}{4} \right)^{5x + 2} = \left( \frac{4}{3} \right)^{2x}
\]

To solve this equation, let's follow these steps:

### Step 1: Rewrite the equation using the property of exponents
We can rewrite \(\left( \frac{4}{3} \right)\) as \(\left( \frac{3}{4} \right)^{-1}\). This gives:

\[
\left( \frac{3}{4} \right)^{5x + 2} = \left( \frac{3}{4} \right)^{-2x}
\]

### Step 2: Set the exponents equal to each other
Since the bases are now the same \(\left( \frac{3}{4} \right)\), we can set the exponents equal to each other:

\[
5x + 2 = -2x
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
5x + 2 = -2x
\]

Add \(2x\) to both sides:

\[
5x + 2x + 2 = 0
\]

\[
7x + 2 = 0
\]

Subtract 2 from both sides:

\[
7x = -2
\]

Divide by 7:

\[
x = -\frac{2}{7}
\]

### Final Answer:
\[
x = -\frac{2}{7}
\]

Would you like any additional explanations or details on this solution?

Here are 5 follow-up questions for you:
1. How would you solve this equation if the bases were not related?
2. Can you explain the property of exponents used here?
3. What happens if the equation involved logarithms instead?
4. How would you check if the solution \(x = -\frac{2}{7}\) is correct?
5. Can you solve a similar equation with different fractions as bases?

**Tip:** When solving exponential equations, it's useful to express both sides with the same base if possible to simplify the comparison of exponents.

Solve the exponential equation (3/4)^(5x + 2) = (4/3)^(2x) with step-by-step solutions. This problem illustrates key concepts of exponents and solving equations by comparing exponents. Suitable for Grades 10-12.

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