https://file.aiquickdraw.com/mathbot/file/1727445069495fzhnw6hb.jpg

Let's go through each of the four problems in the image step by step:

### 1. \( (4^2)^{-2} \)
First, evaluate \( 4^2 \):
\[
4^2 = 16
\]
Now raise it to the power of \( -2 \):
\[
16^{-2} = \frac{1}{16^2} = \frac{1}{256}
\]
Thus, the value is:
\[
\boxed{\frac{1}{256}}
\]

### 2. \( \left( \frac{23}{20} \right)^0 \)
Any non-zero number raised to the power of 0 is 1:
\[
\left( \frac{23}{20} \right)^0 = 1
\]
Thus, the value is:
\[
\boxed{1}
\]

### 3. \( \frac{3^{-2}}{3^2} \)
First, evaluate \( 3^{-2} \) and \( 3^2 \):
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}, \quad 3^2 = 9
\]
Now divide:
\[
\frac{\frac{1}{9}}{9} = \frac{1}{9} \times \frac{1}{9} = \frac{1}{81}
\]
Thus, the value is:
\[
\boxed{\frac{1}{81}}
\]

### 4. \( (2^{-1})^3 \cdot 2^4 \)
First, evaluate \( 2^{-1} \) raised to the power of 3 and \( 2^4 \):
\[
2^{-1} = \frac{1}{2}, \quad \left( \frac{1}{2} \right)^3 = \frac{1}{2^3} = \frac{1}{8}, \quad 2^4 = 16
\]
Now multiply:
\[
\frac{1}{8} \times 16 = 2
\]
Thus, the value is:
\[
\boxed{2}
\]

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Would you like any further details or explanations?

Here are five related questions you could explore:
1. What is the general rule for raising powers to negative exponents?
2. How does the power of zero work for any base?
3. How do you simplify expressions with powers of the same base when dividing?
4. How do negative exponents affect the outcome of a calculation?
5. How does multiplying powers of 2 with different exponents work?

**Tip**: When simplifying powers of the same base, remember that division corresponds to subtracting the exponents!

The user submitted a set of four mathematical problems to evaluate various expressions involving exponents, negative powers, and fractions.

Step-by-step solutions to evaluate expressions with powers, fractions, and negative exponents. Covers exponents, zero power rule, negative exponents, and multiplying powers. Suitable for Grades 7-9.

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