https://file.aiquickdraw.com/mathbot/file/1727373202437fb1kqgvp.jpg

The image contains several problems where the goal is to express each number as a power of two. I'll break down the process and explain the answers for each expression.

### Left Column:
1. **\( 12 - 2^5 \)**
   - \( 12 - 2^5 = 12 - 32 = -20 \).
   This is an expression that results in a negative value, not expressible as a power of 2 directly.

2. **\( \left( \frac{64}{8} \right)^3 \)**
   - First simplify inside the parentheses:
     \( \frac{64}{8} = 8 \).
   - Now raise to the power of 3:
     \( 8^3 = (2^3)^3 = 2^9 \).
   Thus, \( \left( \frac{64}{8} \right)^3 = 2^9 \).

3. **\( 11^2 - 5^4 \)**
   - Calculate each term individually:
     \( 11^2 = 121 \) and \( 5^4 = 625 \).
   - Subtract:
     \( 121 - 625 = -504 \).
   This is a negative result, which cannot be expressed as a power of 2.

4. **\( 12^3 - 4^4 \)**
   - Calculate each term individually:
     \( 12^3 = 1728 \) and \( 4^4 = 256 \).
   - Subtract:
     \( 1728 - 256 = 1472 \).
   Again, this is not directly a power of 2.

5. **\( 4^5 - 2 \)**
   - First calculate \( 4^5 = (2^2)^5 = 2^{10} \).
   - Then subtract 2:
     \( 2^{10} - 2 = 1024 - 2 = 1022 \).
   This is not a power of 2.

### Right Column:
1. **\( 81^2 = 3 \)**
   - Solve for 81 as a power of a smaller base:
     \( 81 = 3^4 \), so \( (3^4)^2 = 3^8 \).
   This expression is incorrect as it cannot equal 3.

2. **\( \left( \frac{1}{2} \right)^5 = 7 \)**
   - \( \left( \frac{1}{2} \right)^5 = 2^{-5} \), but it does not equal 7.

3. **\( \left( \frac{1}{2} \right)^4 = 2 \)**
   - \( \left( \frac{1}{2} \right)^4 = 2^{-4} \), which does not equal 2.

4. **\( \left( \frac{2}{128} \right)^{-3} = 1 \)**
   - Simplify inside the parentheses:
     \( \frac{2}{128} = \frac{1}{64} = 2^{-6} \).
   - Then raise to the power of -3:
     \( \left( 2^{-6} \right)^{-3} = 2^{18} \), which does not equal 1.

5. **\( 256 = 4^4 \)**
   - \( 256 = 2^8 \), but \( 4^4 = (2^2)^4 = 2^8 \), so it is correct in both forms.

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If you want more details or have any questions, feel free to ask! Here are five questions related to this:

1. Can negative results be expressed as powers of 2?
2. What is the simplified form of \( \left( \frac{64}{8} \right)^3 \)?
3. Why are some of the right column equations incorrect?
4. How can \( 256 \) be expressed both as \( 2^8 \) and \( 4^4 \)?
5. Is it possible for a fraction to be expressed as a power of 2?

**Tip**: To express complex expressions as powers of 2, first simplify terms and then look for common bases to convert.

Simplify these expressions to a power of two where possible.

Learn how to express various mathematical expressions as powers of two. This tutorial walks through problems involving exponent rules, powers of fractions, and the simplification of algebraic expressions. Suitable for high school students in Grades 9-12.

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