https://file.aiquickdraw.com/mathbot/file/1725513339334xnp3pu7d.jpg

The equation in the image is:

\[
25^{2x - 1} = 125^{3x + 4}
\]

### Step 1: Express the bases as powers of 5

Since \( 25 = 5^2 \) and \( 125 = 5^3 \), we can rewrite the equation using powers of 5:

\[
(5^2)^{2x - 1} = (5^3)^{3x + 4}
\]

### Step 2: Simplify the exponents

Using the power rule \( (a^m)^n = a^{m \cdot n} \), we simplify both sides:

\[
5^{2(2x - 1)} = 5^{3(3x + 4)}
\]

Simplifying the exponents:

\[
5^{4x - 2} = 5^{9x + 12}
\]

### Step 3: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other:

\[
4x - 2 = 9x + 12
\]

### Step 4: Solve for \( x \)

To solve for \( x \), first move all terms involving \( x \) to one side:

\[
4x - 9x = 12 + 2
\]

\[
-5x = 14
\]

Now, divide by \( -5 \):

\[
x = -\frac{14}{5}
\]

### Final Answer:
\[
x = -\frac{14}{5}
\]

Would you like further details on this process or have any questions?

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Here are 5 related questions to explore further:
1. How would you solve the equation if the bases were not powers of the same number?
2. Can you express other numbers as powers of small primes?
3. What is the general rule for solving exponential equations with the same base?
4. How does changing the base in exponential equations affect the solution process?
5. How would the solution change if the exponents were more complex, e.g., involving higher-degree polynomials?

**Tip**: Always try to express complex numbers in terms of their prime factors when solving exponential equations with different bases.

Solve the equation: 25^(2x - 1) = 125^(3x + 4).

Learn how to solve the exponential equation 25^(2x - 1) = 125^(3x + 4) using properties of exponents and algebraic techniques. This detailed guide walks you through rewriting bases and setting exponents equal, suitable for students in Grades 9-12.

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