To determine the type of number that results from the expression \(512 \cdot \sqrt{16}\), we first evaluate the square root and then perform the multiplication.
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Calculate \( \sqrt{16} \): \[ \sqrt{16} = 4 \]
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Now, substitute back into the expression: \[ 512 \cdot \sqrt{16} = 512 \cdot 4 \]
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Perform the multiplication: \[ 512 \cdot 4 = 2048 \]
Now, let's analyze the result. The number \(2048\) is a whole number, which is rational (since any integer is a rational number).
Thus, the correct response is:
Rational, because both numbers are rational.