What is the product of square root of, 2, end square root 
2
​
  and 5, square root of, 10, end square root5 
10
​
  in simplest radical form?

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!

What is the product of \( \sqrt{2} \) and \( 5\sqrt{10} \) in simplest radical form?

Explore the detailed solution to finding the product of \( \sqrt{2} \) and \( 5\sqrt{10} \), using key mathematical concepts like the multiplication of square roots and radical simplification. Learn the step-by-step process, ideal for students in Grades 8-10.

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