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Let's simplify the given expression step by step:

The expression is:

\[
\sqrt{8y} + 5\sqrt{50y} - 2\sqrt{18y}
\]

### Step 1: Simplify the square roots
We'll simplify each square root term.

1. \(\sqrt{8y} = \sqrt{4 \times 2y} = \sqrt{4} \times \sqrt{2y} = 2\sqrt{2y}\)
2. \(\sqrt{50y} = \sqrt{25 \times 2y} = \sqrt{25} \times \sqrt{2y} = 5\sqrt{2y}\)
3. \(\sqrt{18y} = \sqrt{9 \times 2y} = \sqrt{9} \times \sqrt{2y} = 3\sqrt{2y}\)

### Step 2: Substitute the simplified terms back into the expression
Now, rewrite the expression using the simplified square roots:

\[
2\sqrt{2y} + 5 \times 5\sqrt{2y} - 2 \times 3\sqrt{2y}
\]

### Step 3: Multiply the constants
This becomes:

\[
2\sqrt{2y} + 25\sqrt{2y} - 6\sqrt{2y}
\]

### Step 4: Combine like terms
Now, add and subtract the terms:

\[
(2 + 25 - 6)\sqrt{2y} = 21\sqrt{2y}
\]

### Final Answer:
The simplified expression is:

\[
21\sqrt{2y}
\]

Thus, the correct answer is the first option, **\(21\sqrt{2y}\)**.

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Do you have any questions or would you like more details?

Here are 5 related questions to consider:
1. How do you simplify square roots of larger numbers?
2. What is the process for combining like radical terms?
3. Can you explain why \(\sqrt{8y}\) simplifies to \(2\sqrt{2y}\)?
4. What happens if we have different square root terms, such as \(\sqrt{y}\) and \(\sqrt{2y}\)?
5. How would this change if we had cube roots instead of square roots?

**Tip:** Always look for perfect square factors when simplifying square roots. This helps in reducing terms easily.

Simplify the expression: √(8y) + 5√(50y) - 2√(18y)

Simplify the expression √(8y) + 5√(50y) - 2√(18y) using square root properties and combine like terms. This algebra problem walks you through radical expression simplification, perfect square factors, and final answers with a detailed solution. Suitable for high school students in Grades 9-12.

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