To multiply \( 9\sqrt{3} \) and \( 6\sqrt{7} \), you can multiply the coefficients (the numbers outside the square roots) and the square roots separately.
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Multiply the coefficients: \[ 9 \times 6 = 54 \]
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Multiply the square roots: \[ \sqrt{3} \times \sqrt{7} = \sqrt{3 \times 7} = \sqrt{21} \]
Now combine the results: \[ 9\sqrt{3} \cdot 6\sqrt{7} = 54\sqrt{21} \]
Thus, the exact answer is: \[ \boxed{54\sqrt{21}} \]