To simplify the expression $3\sqrt{567}$, we need to first simplify the square root of 567.

Step 1: Factorize 567

We start by finding the prime factorization of 567: $567 \div 3 = 189 \quad (3 \text{ is a prime factor})$ $189 \div 3 = 63 \quad (3 \text{ is a prime factor})$ $63 \div 3 = 21 \quad (3 \text{ is a prime factor})$ $21 \div 3 = 7 \quad (7 \text{ is a prime factor})$ So, the prime factorization of 567 is: $567 = 3^4 \times 7$

Step 2: Simplify the Square Root

Now, simplify the square root: $\sqrt{567} = \sqrt{3^4 \times 7} = \sqrt{(3^2 \times 3^2) \times 7} = 3^2 \times \sqrt{7} = 9\sqrt{7}$

Step 3: Multiply by 3

Finally, multiply by the coefficient 3: $3\sqrt{567} = 3 \times 9\sqrt{7} = 27\sqrt{7}$

Final Answer:

$3\sqrt{567} = 27\sqrt{7}$

Would you like more details on this process or have any questions?

Here are 5 related questions you might find useful:

1. How do you simplify $2\sqrt{72}$?
2. What is the simplified form of $4\sqrt{200}$?
3. Can you simplify the expression $5\sqrt{180}$?
4. How would you simplify $6\sqrt{288}$?
5. What is the simplified form of $7\sqrt{98}$?

Tip: When simplifying expressions with radicals, it's often easier to simplify the radical first and then multiply by any coefficients outside the square root.