To find the value of the expression \(2^2 \cdot 3^3\), we need to calculate each part separately:
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Calculate \(2^2\): \[ 2^2 = 4 \]
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Calculate \(3^3\): \[ 3^3 = 27 \]
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Now multiply the results: \[ 4 \cdot 27 = 108 \]
Therefore, the whole number that the product \(2^2 \cdot 3^3\) represents is \( \boxed{108} \).