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We are asked to find the product of \((3\sqrt{2})(5\sqrt{10})\).

### Step-by-Step Solution:
1. Multiply the whole numbers: \( 3 \times 5 = 15 \).
2. Multiply the square roots: \(\sqrt{2} \times \sqrt{10} = \sqrt{2 \times 10} = \sqrt{20}\).

Thus, the product is:
\[
(3\sqrt{2})(5\sqrt{10}) = 15\sqrt{20}
\]

3. Simplify \(\sqrt{20}\):
\[
\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}
\]

4. Multiply this by 15:
\[
15\sqrt{20} = 15 \times 2\sqrt{5} = 30\sqrt{5}
\]

So, the final answer is:
\[
30\sqrt{5}
\]

Would you like further details or have any questions?

Here are five related questions to extend your understanding:
1. How do you multiply two square roots like \(\sqrt{a} \times \sqrt{b}\)?
2. Can all square roots be simplified? Why or why not?
3. What is the result of squaring a square root, like \((\sqrt{5})^2\)?
4. How do you simplify expressions involving square roots and whole numbers together?
5. What are some real-world applications where square roots are used?

**Tip**: When simplifying square roots, always look for perfect squares inside the radical (like \(\sqrt{4}\) or \(\sqrt{9}\)) to simplify your expression further.

Learn how to multiply two radicals: (3√2)(5√10). This step-by-step solution explains how to multiply the numbers and simplify square roots, perfect for students in Grades 9-10.

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