To find the product of \( 3\sqrt{3} \) and \( 3\sqrt{30} \), we can multiply them together using the properties of multiplication and radicals.
\[ 3\sqrt{3} \times 3\sqrt{30} = (3 \times 3) \times (\sqrt{3} \times \sqrt{30}) \]
First, we multiply \( 3 \times 3 \):
\[ 3 \times 3 = 9 \]
Next, we multiply the square roots:
\[ \sqrt{3} \times \sqrt{30} = \sqrt{3 \times 30} = \sqrt{90} \]
Now we combine these results:
\[ 9 \times \sqrt{90} \]
Next, we simplify \( \sqrt{90} \). We can factor \( 90 \) into its prime factors:
\[ 90 = 9 \times 10 = 3^2 \times 10 \]
The square root of \( 90 \) can be simplified as follows:
\[ \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10} \]
Now we can substitute this back into our product:
\[ 9 \times \sqrt{90} = 9 \times 3\sqrt{10} = 27\sqrt{10} \]
Therefore, the product of \( 3\sqrt{3} \) and \( 3\sqrt{30} \) in simplest radical form is
\[ \boxed{27\sqrt{10}} \]
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