The image contains a mathematical expression along with multiple-choice options to find its value. Let's break down the expression and solve it.

Expression:

$\frac{4^2 \cdot 4^{-2}}{4^4 \div 4^4}$

Simplifying the Expression:

1. Numerator: $4^2 \cdot 4^{-2}$

   • Using the property of exponents: $a^m \cdot a^n = a^{m+n}$
   • $4^2 \cdot 4^{-2} = 4^{2 + (-2)} = 4^0 = 1$

2. Denominator: $4^4 \div 4^4$

   • Using the property of exponents: $\frac{a^m}{a^n} = a^{m-n}$
   • $4^4 \div 4^4 = 4^{4-4} = 4^0 = 1$

3. Final Expression: $\frac{1}{1} = 1$

Conclusion:

The value of the expression is $1$.

Correct Option:

The correct answer is:

• Option 3: $1$

Would you like further details on the solution or have any questions? Here are five related questions for practice:

1. How would the expression change if the numerator were $4^3 \cdot 4^{-1}$?
2. What is the value of $4^{-2}$ and how does it affect the multiplication in the numerator?
3. Can you simplify $4^4 \div 4^{-2}$ and what would the result be?
4. What is the general rule for multiplying and dividing powers with the same base?
5. How would the expression change if the entire numerator was squared?

Tip: Remember, $a^0 = 1$ for any non-zero value of $a$. This property is very useful in simplifying expressions involving exponents.