https://file.aiquickdraw.com/mathbot/file/1729176572487ohjb3kwj.jpg

The equation provided in the image is:

\[
\left( \frac{1}{4} \right)^{2x+1} = 32
\]

Let's solve for \( x \).

### Step 1: Rewrite the equation using powers of 2

We know that:
\[
\frac{1}{4} = 4^{-1} = (2^2)^{-1} = 2^{-2}
\]
Thus, the left-hand side becomes:
\[
(2^{-2})^{2x+1} = 32
\]
Simplifying the exponent:
\[
2^{-2(2x+1)} = 32
\]
\[
2^{-4x-2} = 32
\]

### Step 2: Express \( 32 \) as a power of 2
\[
32 = 2^5
\]

### Step 3: Set the exponents equal to each other
Since the bases are the same (both are powers of 2), we can equate the exponents:
\[
-4x - 2 = 5
\]

### Step 4: Solve for \( x \)
\[
-4x = 5 + 2
\]
\[
-4x = 7
\]
\[
x = -\frac{7}{4}
\]

This value is not listed in the options, so let's revisit the problem and check the base. There might be an interpretation issue or error in how the problem was presented. Do you need further clarification on this?

---

Let me know if you have questions or if you'd like more details. 

Here are 5 follow-up questions to expand your understanding:
1. How do you convert a fraction to a power of 2?
2. What is the role of exponent rules in solving such equations?
3. Can you find other examples where powers of fractions need to be simplified?
4. How do you handle negative exponents in equations?
5. How can you apply this technique to other bases, such as 3 or 5?

**Tip:** When dealing with exponents, always try to express both sides of the equation with the same base. This simplifies the process greatly!

What is the solution to the equation (1/4)^(2x+1) = 32?

Learn how to solve the exponential equation (1/4)^(2x+1) = 32 step-by-step. Understand the use of exponent properties and techniques for equating exponents. This example is suitable for students in Grades 9-11 studying algebra and exponential equations.

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