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.

Learn how to simplify the expression 3√567 step-by-step. This problem involves radicals and prime factorization, providing a clear method to simplify radicals. Suitable for Grades 7-9.

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