https://file.aiquickdraw.com/mathbot/file/1730334899878bclmxyag.jpg

The equation shown in the image is:

\[
4^{x+1} = \frac{1}{64}
\]

To solve for \( x \), we can rewrite both sides of the equation with the same base. Here’s the step-by-step solution:

### Step 1: Rewrite \( \frac{1}{64} \) as a power of 4
Since \( 64 = 4^3 \), we can write \( \frac{1}{64} \) as:
\[
\frac{1}{4^3} = 4^{-3}
\]

So the equation becomes:
\[
4^{x+1} = 4^{-3}
\]

### Step 2: Set the exponents equal to each other
Since the bases are the same, we can equate the exponents:
\[
x + 1 = -3
\]

### Step 3: Solve for \( x \)
Subtract 1 from both sides:
\[
x = -3 - 1
\]
\[
x = -4
\]

### Final Answer
\[
x = -4
\]

Would you like further details or have any questions about the steps?

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Here are five related questions to explore this topic further:

1. What is an exponential equation, and how do we solve equations with different bases?
2. How can we handle exponential equations where the bases cannot be easily matched?
3. What is the meaning of a negative exponent, and why does \(4^{-3} = \frac{1}{64}\)?
4. How do we approach solving equations with logarithmic properties?
5. How would the solution change if the base were different (e.g., \(3^{x+1} = \frac{1}{27}\))?

**Tip:** When solving exponential equations, always check if you can rewrite both sides with the same base—it often simplifies the problem significantly!

Learn how to solve the exponential equation 4^(x+1) = 1/64 by rewriting the equation with the same base. This solution covers exponent properties and is suitable for students in Grades 9-11.

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