To multiply and simplify the expression $\sqrt[3]{7} \times \sqrt{3}$, follow these steps:

Step 1: Express each radical

• The first part is $\sqrt[3]{7}$, which is the cube root of 7.
• The second part is $\sqrt{3}$, which is the square root of 3.

Step 2: Multiply the terms

To multiply radicals with different indices, you can multiply the numbers under the radicals first:

[ \sqrt[3]{7} \times \sqrt{3} = \sqrt[3]{7} \times \sqrt[3]{3^{3/2}} = \sqrt[3]{7 \cdot 3^{3/2}}. ]

Step 3: Simplify

There is no straightforward simplification without a numerical approximation, so the answer remains:

$\boxed{\sqrt[3]{7 \cdot 3^{3/2}}}.$

If you wish to proceed with an approximation, that can be done as well.

Would you like further explanation or a numerical estimate?