Let's simplify the expression \( 5\sqrt{18} - 2\sqrt{8} - 10\sqrt{200} \) step by step.

### Step 1: Simplify the square roots

1. \( \sqrt{18} \) can be simplified as:
   \[
   \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}
   \]

2. \( \sqrt{8} \) can be simplified as:
   \[
   \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
   \]

3. \( \sqrt{200} \) can be simplified as:
   \[
   \sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}
   \]

### Step 2: Substitute these into the original expression

Now we can rewrite the expression:
\[
5\sqrt{18} - 2\sqrt{8} - 10\sqrt{200} = 5(3\sqrt{2}) - 2(2\sqrt{2}) - 10(10\sqrt{2})
\]
\[
= 15\sqrt{2} - 4\sqrt{2} - 100\sqrt{2}
\]

### Step 3: Combine like terms

Since all terms involve \( \sqrt{2} \), we can combine them:
\[
15\sqrt{2} - 4\sqrt{2} - 100\sqrt{2} = (15 - 4 - 100)\sqrt{2} = -89\sqrt{2}
\]

### Final Answer
The simplified form of the expression is:
\[
-89\sqrt{2}
\]

Would you like further details or have any questions? Here are five related questions you might consider:

1. How do you simplify other square roots like \( \sqrt{72} \) or \( \sqrt{50} \)?
2. What are the steps to rationalize a denominator containing square roots?
3. How do you add or subtract expressions involving different square roots?
4. Can square roots of negative numbers be simplified similarly? What about complex numbers?
5. How does the distributive property apply when simplifying square root expressions?

**Tip:** Always look for the largest perfect square factor when simplifying square roots to make the process easier!

Learn how to simplify the expression 5√18 - 2√8 - 10√200 with a step-by-step solution. This problem covers key concepts such as simplifying square roots and combining like terms, suitable for students in Grades 8-10.

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