To simplify the expression \( 10\sqrt{45} - \sqrt{20} \), we can break down each square root by factoring out perfect squares.

1. **Simplify \( \sqrt{45} \):**
   \[
   \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}
   \]
   So, \( 10\sqrt{45} = 10 \times 3\sqrt{5} = 30\sqrt{5} \).

2. **Simplify \( \sqrt{20} \):**
   \[
   \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}
   \]
   So, \( \sqrt{20} = 2\sqrt{5} \).

Now substitute the simplified forms back into the original expression:
\[
10\sqrt{45} - \sqrt{20} = 30\sqrt{5} - 2\sqrt{5}
\]

3. **Combine like terms**:
\[
30\sqrt{5} - 2\sqrt{5} = (30 - 2)\sqrt{5} = 28\sqrt{5}
\]

Thus, the simplified expression is:
\[
28\sqrt{5}
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you simplify expressions with square roots involving variables?
2. What are the properties of square roots and how do they apply in algebra?
3. How can you simplify square roots of larger numbers?
4. What are perfect squares and how do they help in simplifying radicals?
5. How do you multiply and divide expressions with square roots?

**Tip**: When simplifying square roots, always look for the largest perfect square factor of the number under the root.

This step-by-step solution explains how to simplify the expression 10√45 - √20 using properties of square roots and combining like terms. Suitable for students in Grades 9-11, this problem reinforces understanding of radicals and algebraic simplification.

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