To simplify the expression \( 5\sqrt{63} - 2\sqrt{175} + \sqrt{252} \), let's break down each square root by factoring out the perfect squares.

1. **Simplify \( 5\sqrt{63} \):**
   \[
   63 = 9 \times 7 \quad \text{and} \quad \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}
   \]
   So,
   \[
   5\sqrt{63} = 5 \times 3\sqrt{7} = 15\sqrt{7}
   \]

2. **Simplify \( -2\sqrt{175} \):**
   \[
   175 = 25 \times 7 \quad \text{and} \quad \sqrt{175} = \sqrt{25 \times 7} = \sqrt{25} \times \sqrt{7} = 5\sqrt{7}
   \]
   So,
   \[
   -2\sqrt{175} = -2 \times 5\sqrt{7} = -10\sqrt{7}
   \]

3. **Simplify \( \sqrt{252} \):**
   \[
   252 = 36 \times 7 \quad \text{and} \quad \sqrt{252} = \sqrt{36 \times 7} = \sqrt{36} \times \sqrt{7} = 6\sqrt{7}
   \]

Now, combine all the simplified terms:
\[
15\sqrt{7} - 10\sqrt{7} + 6\sqrt{7}
\]

Add the coefficients of \( \sqrt{7} \):
\[
(15 - 10 + 6)\sqrt{7} = 11\sqrt{7}
\]

Thus, the simplified expression is \( \boxed{11\sqrt{7}} \).

Would you like further details or have any questions?

Here are five related questions:
1. How do you factor square roots to simplify radical expressions?
2. What are the steps to add and subtract like radicals?
3. How do you handle square roots of non-perfect squares?
4. What methods can be used to simplify expressions involving multiple radicals?
5. How does simplifying radicals relate to rationalizing the denominator?

**Tip:** Always look for the largest perfect square factor when simplifying square roots!

Learn how to simplify the expression \( 5\sqrt{63} - 2\sqrt{175} + \sqrt{252} \) by factoring out perfect squares and combining like terms. This problem involves simplifying radical expressions and is suitable for Grades 9-12 students.

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