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.

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.