To generate equivalent expressions for \( 2^2 \cdot 2^{-4} \) using properties of integer exponents, we can use the property that states \( a^m \cdot a^n = a^{m+n} \).
So we can combine the exponents:
\[ 2^2 \cdot 2^{-4} = 2^{2 + (-4)} = 2^{-2} \]
Next, we want to express \( 2^{-2} \) with only positive exponents. We can rewrite \( 2^{-2} \) as:
\[ 2^{-2} = \frac{1}{2^2} \]
Now, we can simplify \( \frac{1}{2^2} \):
\[ \frac{1}{2^2} = \frac{1}{4} \]
To provide a whole number, we see that \( \frac{1}{4} \) is not a whole number but we can express our result in decimal form or as a whole number fraction.
Thus, \( 2^{2} \cdot 2^{-4} = \frac{1}{4} \).
The final answer is:
\[ \frac{1}{4} \quad \text{(as a fraction)} \]
However, since the request is for a whole number, we state that the expression \( 2^2 \cdot 2^{-4} = \frac{1}{4} \) does not yield a whole number solution. If strictly required to provide a whole number response, you would specify that the result is \( 0 \) (not whole, but the closest under integer requirements).
If expressing as a whole number is essential, acknowledge its fractional part, otherwise state directly \( \frac{1}{4} \) is the most reduced expression.
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