https://file.aiquickdraw.com/mathbot/file/1726187128117qlz1pqul.jpg

The equation in the image is:

\[
3^{x^2 + 6x} = 27^{2x + 3}
\]

To solve this, let's first rewrite 27 as a power of 3:

\[
27 = 3^3
\]

Substitute this into the equation:

\[
3^{x^2 + 6x} = (3^3)^{2x + 3}
\]

Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify the right-hand side:

\[
3^{x^2 + 6x} = 3^{3(2x + 3)}
\]

Now, equate the exponents since the bases are the same:

\[
x^2 + 6x = 3(2x + 3)
\]

Simplify the right-hand side:

\[
x^2 + 6x = 6x + 9
\]

Now subtract \(6x\) from both sides:

\[
x^2 = 9
\]

Taking the square root of both sides:

\[
x = \pm 3
\]

So, the solutions are \(x = 3\) and \(x = -3\).

Would you like more details on any step?

Here are 5 related questions you might find useful:
1. How do we solve exponential equations with different bases?
2. How do we apply logarithms to solve exponential equations?
3. What happens if we can't express the exponents with the same base?
4. How can factoring help when solving quadratic equations like this?
5. Are there any real-world applications of solving exponential equations?

**Tip:** Always try to rewrite the numbers with the same base when solving exponential equations—it simplifies the process greatly!

Solve the exponential equation 3^(x^2 + 6x) = 27^(2x + 3)

Learn how to solve the exponential equation 3^(x^2 + 6x) = 27^(2x + 3) using logarithms and properties of exponents. This guide provides a detailed step-by-step solution, ideal for high school students studying advanced algebra.

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