Let's analyze the options to determine which expression is a perfect square.

1. Option 1: $256z^4$

   • $256$ is a perfect square because $256 = 16^2$.
   • $z^4$ is also a perfect square because $z^4 = (z^2)^2$.
   • Therefore, $256z^4$ is a perfect square.

2. Option 2: $25x$

   • $25$ is a perfect square because $25 = 5^2$, but $x$ alone is not necessarily a perfect square unless more information about $x$ is given. Therefore, $25x$ is not guaranteed to be a perfect square.

3. Option 3: $66$

   • $66$ is not a perfect square because no integer squared gives 66.

4. Option 4: $110$

   • $110$ is not a perfect square for the same reason as 66; no integer squared gives 110.

Conclusion:

The correct answer is $256z^4$, which is a perfect square.

Would you like more details or have any questions about this solution?
Here are five related questions:

1. How do you determine if a number is a perfect square?
2. Can a variable expression, like $z^4$, always be considered a perfect square?
3. Why is $256$ a perfect square but $66$ is not?
4. How can we generalize the process of finding perfect squares in algebraic expressions?
5. What role do exponents play in determining whether an expression is a perfect square?

Tip: A number is a perfect square if its prime factors all have even exponents.