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The given problem asks to simplify the square root of 192 step by step.

To reduce \(\sqrt{192}\), we need to factor 192 into its prime factors and find the perfect squares.

### Step-by-Step Simplification:
1. Start by finding the prime factorization of 192.
   
   \(192 \div 2 = 96\)  
   \(96 \div 2 = 48\)  
   \(48 \div 2 = 24\)  
   \(24 \div 2 = 12\)  
   \(12 \div 2 = 6\)  
   \(6 \div 2 = 3\)  
   \(3 \div 3 = 1\)
   
   So, the prime factorization of 192 is:
   \[
   192 = 2^6 \times 3
   \]

2. Group the factors into perfect squares where possible:
   \[
   192 = (2^6) \times 3 = (2^3)^2 \times 3
   \]

3. Simplify the square root by taking out the perfect square:
   \[
   \sqrt{192} = \sqrt{(2^3)^2 \times 3} = 2^3 \times \sqrt{3} = 8\sqrt{3}
   \]

### Final Answer:
\[
\sqrt{192} = 8\sqrt{3}
\]

Would you like more details on any of the steps or further clarification?

### Related Questions:
1. How do you factor other square roots into prime factors?
2. What is the method to handle cube roots or higher roots similarly?
3. How do you simplify \(\sqrt{75}\) using prime factorization?
4. What happens when the number inside the radical is already a perfect square?
5. How can you break down cube roots using prime factorization?

### Tip:
Always check if the number inside the square root has any perfect square factors, as this will help simplify the radical easily!

Go step by step to reduce the radical. \( \sqrt{192} \)

Learn how to simplify \( \sqrt{192} \) step by step using prime factorization. Follow this guide to reduce radicals, find perfect squares, and simplify expressions. Suitable for students in Grades 8-10.

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