https://file.aiquickdraw.com/mathbot/file/1726513396149eqlzvxn3.jpg

The equation in the image is:

\[
5^{3x + 7} = 25^{x - 4}
\]

We can solve this equation by expressing both sides with the same base. Notice that \(25 = 5^2\), so we rewrite the right-hand side:

\[
5^{3x + 7} = (5^2)^{x - 4}
\]

Now, simplify the right-hand side:

\[
5^{3x + 7} = 5^{2(x - 4)} = 5^{2x - 8}
\]

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

\[
3x + 7 = 2x - 8
\]

Now, solve for \(x\):

\[
3x - 2x = -8 - 7
\]
\[
x = -15
\]

Thus, the solution is:

\[
x = -15
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How can you solve exponential equations when the bases are different?
2. What are the steps for solving logarithmic equations?
3. How do you handle equations with variables in the exponent if the bases can't be easily matched?
4. Can you solve for \(x\) in \(7^{x+1} = 49^{2x}\)?
5. How does changing the base affect the complexity of solving exponential equations?

**Tip**: When solving exponential equations, always try to express both sides with the same base, as it simplifies the process.

Solve the equation: 5^(3x + 7) = 25^(x - 4).

Learn how to solve the exponential equation 5^(3x + 7) = 25^(x - 4) by simplifying both sides to the same base and equating the exponents. This detailed step-by-step solution covers key algebraic concepts for high school students (Grades 9-12).

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