To find the product of $\sqrt{2}$ and $5\sqrt{10}$, we multiply the constants and the radicals separately.

The product is: $\sqrt{2} \times 5\sqrt{10}$ This can be rewritten as: $5 \times (\sqrt{2} \times \sqrt{10})$ Using the property of square roots $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, we have: $\sqrt{2} \times \sqrt{10} = \sqrt{2 \times 10} = \sqrt{20}$ Thus, the expression becomes: $5 \times \sqrt{20}$ Now simplify $\sqrt{20}$. Since $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$, the expression becomes: $5 \times 2\sqrt{5} = 10\sqrt{5}$ So, the product of $\sqrt{2}$ and $5\sqrt{10}$ in simplest radical form is: $10\sqrt{5}$

Would you like further explanation or have any other questions?

Here are 5 questions to explore this further:

1. How would you multiply $\sqrt{3}$ by $4\sqrt{12}$?
2. What is the product of $2\sqrt{7}$ and $3\sqrt{14}$?
3. Can you simplify $\sqrt{8} \times \sqrt{18}$?
4. How do you rationalize the denominator of $\frac{1}{\sqrt{2}}$?
5. How would you simplify $\sqrt{50}$?

Tip: When multiplying square roots, combine the numbers under the radicals first, then simplify!