https://file.aiquickdraw.com/mathbot/file/1728696600332fbpvrkxv.jpg

The equation given in the image is:

\[
(6^n)^4 = 6^{-12}
\]

We can solve this by simplifying both sides. Using the rule of exponents \((a^m)^n = a^{m \cdot n}\), the left side becomes:

\[
(6^n)^4 = 6^{4n}
\]

Now, the equation is:

\[
6^{4n} = 6^{-12}
\]

Since the bases (6) are the same, we can set the exponents equal to each other:

\[
4n = -12
\]

Solving for \(n\):

\[
n = \frac{-12}{4} = -3
\]

Thus, the value of \(n\) is:

\[
\boxed{-3}
\]

Would you like further details or explanations?

Here are five related questions to expand on this:

1. What happens when the exponent of a number is zero?
2. How do you solve equations with fractional exponents?
3. What is the difference between a positive and negative exponent?
4. Can you apply the same exponent rules to variables instead of numbers?
5. How do you simplify expressions with multiple exponent rules applied?

**Tip:** When dealing with exponents, always simplify the powers first and check if the bases are the same. This makes it easier to compare exponents.

(6^n)^4 = 6^{-12}. What is the value of n?

Learn how to solve the equation (6^n)^4 = 6^{-12} by applying exponent rules. This detailed step-by-step solution is suitable for students in Grades 8-10 and covers important concepts such as exponents, the equality of exponents, and exponent multiplication.

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