To multiply \( 9\sqrt{3} \) and \( 6\sqrt{7} \), we can multiply the coefficients 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, put them together: \[ 9\sqrt{3} \times 6\sqrt{7} = 54\sqrt{21} \]
So, the final answer is: \[ 54\sqrt{21} \]