https://file.aiquickdraw.com/mathbot/file/1727023687134lb0ade2b.jpg

We are given the equation:

\[
8^c = (8^{-4})^5
\]

To find the value of \( c \), let's simplify the right-hand side.

### Step 1: Simplify the right-hand side
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\), we can simplify \((8^{-4})^5\):

\[
(8^{-4})^5 = 8^{-4 \cdot 5} = 8^{-20}
\]

### Step 2: Set the exponents equal
Now we have the equation:

\[
8^c = 8^{-20}
\]

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

\[
c = -20
\]

Thus, the value of \( c \) is \(-20\).

Would you like further details or clarifications?

### Related questions:
1. How do we use the power of a power rule in exponents?
2. What happens when bases are the same on both sides of an equation?
3. How would the solution change if the base was different from 8?
4. How can we solve exponent equations with variables on both sides?
5. Can you explain how negative exponents work in other cases?

### Tip:
Always look for common bases when dealing with exponent equations; this allows you to simplify the expressions and solve for unknown variables efficiently.

In the following equation, what is the value of c?
8^c = (8^{-4})^5

Learn how to solve the exponent equation 8^c = (8^{-4})^5 by applying the power of a power rule. This step-by-step solution covers key concepts like simplifying exponents and setting equal bases. Suitable for students in Grades 9-11.

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