https://file.aiquickdraw.com/mathbot/file/1728517539657mpg4f24d.jpg

The equation given in the image is:

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

### Step 1: Rewrite the fraction as a power of 4

We know that \( 64 = 4^3 \), so:

\[
\frac{1}{64} = \frac{1}{4^3} = 4^{-3}
\]

Thus, the equation becomes:

\[
4^{x+1} = 4^{-3}
\]

### Step 2: Equating exponents

Since the bases are the same (both are powers of 4), we can set the exponents equal to each other:

\[
x + 1 = -3
\]

### Step 3: Solve for \(x\)

Now, solve for \(x\):

\[
x = -3 - 1 = -4
\]

### Final Answer:

\[
x = -4
\]

Would you like further explanation on any step? Here are some related questions you might consider:

1. How can you express numbers as powers of different bases?
2. What are other examples where equating exponents is useful?
3. How can you solve exponential equations with different bases?
4. What is the general approach for solving \(a^{x} = a^{y}\)?
5. Can this approach be applied to logarithmic equations?

**Tip:** When solving exponential equations, it's helpful to express all terms with the same base, making it easier to compare exponents.

Learn how to solve the exponential equation 4^(x+1) = 1/64. This step-by-step solution covers key concepts like rewriting fractions as powers and equating exponents. Suitable for students in Grades 9-12.

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