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### Problem:
Find the product of \( 5\sqrt{2} \) and \( 2\sqrt{3} \) in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

### Step 1: Multiply the Radicals
To find the product of \( 5\sqrt{2} \) and \( 2\sqrt{3} \), multiply the coefficients (5 and 2) and then multiply the square roots.

\[
5\sqrt{2} \times 2\sqrt{3} = (5 \times 2)(\sqrt{2} \times \sqrt{3}) = 10\sqrt{6}
\]

### Step 2: Simplify the Result
The expression \( 10\sqrt{6} \) is already in its simplest form because \( \sqrt{6} \) cannot be simplified further.

### Step 3: Determine if the Result is Rational or Irrational
A number is rational if it can be expressed as the ratio of two integers. The square root of a non-perfect square (like 6) is irrational. Therefore, \( \sqrt{6} \) is irrational, and any non-zero multiple of an irrational number is also irrational.

So, \( 10\sqrt{6} \) is **irrational**.

### Final Answer:
- **Result:** \( 10\sqrt{6} \)
- **Rational or Irrational:** Irrational
- **Explanation:** The result is irrational because it involves the square root of a non-perfect square (6), which cannot be written as a ratio of two integers and has a non-terminating, non-repeating decimal expansion.

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. What is the product of \( 3\sqrt{5} \) and \( 4\sqrt{2} \)?
2. How do you determine if a number involving a square root is rational or irrational?
3. Can the product of two irrational numbers be rational? If so, give an example.
4. What are the properties of square roots when multiplied together?
5. How do you simplify the product of two different square roots?

**Tip:** When multiplying square roots, always check if the product can be simplified further by looking for perfect squares under the radical.

Find the product of 5√2 and 2√3 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

Learn how to find the product of radicals 5√2 and 2√3 in simplest form and determine if the result is rational or irrational. This problem covers key concepts in algebra, including multiplication of radicals and the properties of irrational numbers. Suitable for students in Grades 9-10.

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