What is the product of 9, square root of, 3, end square root 93​ and 5, square root of, 21, end square root 521​ in simplest radical form?

To find the product of \( 9\sqrt{3} \) and \( 5\sqrt{21} \) in simplest radical form, follow these steps:

1. **Multiply the coefficients**: Multiply the numbers outside the square roots.
   \[
   9 \times 5 = 45
   \]

2. **Multiply the radicands**: Multiply the numbers inside the square roots.
   \[
   \sqrt{3} \times \sqrt{21} = \sqrt{3 \times 21} = \sqrt{63}
   \]

3. **Simplify the square root, if possible**: Break down \( \sqrt{63} \) into its prime factors.
   \[
   \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}
   \]

4. **Combine the results**: Now multiply the result from step 1 with the simplified radical from step 3.
   \[
   45 \times 3\sqrt{7} = 135\sqrt{7}
   \]

So, the product of \( 9\sqrt{3} \) and \( 5\sqrt{21} \) in simplest radical form is \( \boxed{135\sqrt{7}} \).

Would you like any further details or explanations? 

Here are some related questions:
1. How do you simplify expressions involving square roots?
2. What are the properties of square roots used in this problem?
3. How can you determine if a square root can be simplified?
4. What is the process for multiplying and simplifying radical expressions?
5. What happens when you multiply radicals with different indices?

**Tip:** Always check if the radicand can be simplified further by factoring it into squares.

Learn how to calculate the product of 9√3 and 5√21 in simplest radical form with a step-by-step solution. This problem covers radicals and their multiplication, including simplifying square roots. Suitable for students in Grades 9-12.

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