https://file.aiquickdraw.com/mathbot/file/1726667694896so2b9uwz.jpg

It seems that the text extraction from the image was not successful. However, based on the image you uploaded, I can read the equations manually. Here are the problems:

1. \( 2^x = 32 \)
2. \( 3^x = \frac{1}{9} \)
3. \( 2^{x+1} = 64 \)
4. \( \left(\frac{1}{3}\right)^x = 27 \)
5. \( 4^x = \frac{1}{2} \)

Let me solve these one by one.

### 1. \( 2^x = 32 \)
We know \( 32 = 2^5 \), so:
\[
2^x = 2^5 \implies x = 5
\]

### 2. \( 3^x = \frac{1}{9} \)
We can rewrite \( \frac{1}{9} \) as \( 9^{-1} \) and \( 9 = 3^2 \), so:
\[
3^x = 3^{-2} \implies x = -2
\]

### 3. \( 2^{x+1} = 64 \)
We know \( 64 = 2^6 \), so:
\[
2^{x+1} = 2^6 \implies x+1 = 6 \implies x = 5
\]

### 4. \( \left(\frac{1}{3}\right)^x = 27 \)
We know \( \frac{1}{3} = 3^{-1} \) and \( 27 = 3^3 \), so:
\[
(3^{-1})^x = 3^3 \implies 3^{-x} = 3^3 \implies -x = 3 \implies x = -3
\]

### 5. \( 4^x = \frac{1}{2} \)
We know \( 4 = 2^2 \), so we rewrite the equation as:
\[
(2^2)^x = \frac{1}{2} \implies 2^{2x} = 2^{-1} \implies 2x = -1 \implies x = -\frac{1}{2}
\]

---

Here are the solutions:
1. \( x = 5 \)
2. \( x = -2 \)
3. \( x = 5 \)
4. \( x = -3 \)
5. \( x = -\frac{1}{2} \)

Do you need more details on any of these? 

Here are 5 questions to expand:
1. How can you solve exponential equations involving logarithms?
2. What is the relationship between negative exponents and reciprocals?
3. How do you solve exponential equations when bases are not the same?
4. Can you explain how to simplify expressions with fractional exponents?
5. What are the properties of exponents used in solving these equations?

**Tip**: Always express numbers with the same base when solving exponential equations—it simplifies the comparison and solution.

Solve the following exponential equations: 1. 2^x = 32, 2. 3^x = 1/9, 3. 2^(x+1) = 64, 4. (1/3)^x = 27, 5. 4^x = 1/2.

This guide provides solutions to exponential equations such as 2^x = 32, 3^x = 1/9, and others. It covers key concepts like exponents and their properties, and uses logarithms for solving when bases are different. Suitable for high school students.

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