https://file.aiquickdraw.com/mathbot/file/1729356995569e77huftc.jpg

The equation in the image is:

\[
9 \cdot 3^{7x} = (1^2)^{x}
\]

Let's break it down step-by-step:

1. The right-hand side simplifies as \( (1^2)^x = 1 \), since \(1^2 = 1\) and any power of 1 is still 1. So, the equation becomes:
   \[
   9 \cdot 3^{7x} = 1
   \]

2. Next, we rewrite \(9\) as \(3^2\), so the equation becomes:
   \[
   3^2 \cdot 3^{7x} = 1
   \]

3. Now, combine the powers of 3 using the property of exponents \(a^m \cdot a^n = a^{m+n}\):
   \[
   3^{2 + 7x} = 1
   \]

4. Since \(3^{2 + 7x} = 1\), we know that for any non-zero base, \(a^0 = 1\). Therefore, the exponent must be equal to 0:
   \[
   2 + 7x = 0
   \]

5. Solving for \(x\):
   \[
   7x = -2
   \]
   \[
   x = -\frac{2}{7}
   \]

Thus, the solution to the equation is:
\[
x = -\frac{2}{7}
\]

Would you like further clarification on any part of the solution? Here are some related questions to think about:

1. How would the solution change if the base of the exponent was different from 3?
2. What would happen if the equation was \(9 \cdot 3^{7x} = 3^{2x}\)?
3. Can you use logarithms to solve this equation instead of using properties of exponents?
4. What if the right-hand side was \(2^x\) instead of 1?
5. How can you verify the solution \(x = -\frac{2}{7}\)?

**Tip**: When simplifying exponential equations, always try to express all terms with the same base to apply the properties of exponents easily.

Solve the equation analytically: 9 * 3^(7x) = (1^2)^x

Learn how to solve the exponential equation 9 * 3^(7x) = (1^2)^x with a detailed step-by-step solution. This problem covers key algebraic concepts, including the properties of exponents and simplification techniques. Suitable for students in Grades 9-12.

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