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The problem asks whether the equation \(\sqrt{a} - \sqrt{b} = \sqrt{a - b}\) is true or false for \(a, b > 0\).

Let's analyze this equation step by step.

### Consider \(a = 4\) and \(b = 1\):
- \(\sqrt{a} = \sqrt{4} = 2\)
- \(\sqrt{b} = \sqrt{1} = 1\)
- \(\sqrt{a} - \sqrt{b} = 2 - 1 = 1\)
- \(a - b = 4 - 1 = 3\), so \(\sqrt{a - b} = \sqrt{3}\).

Here, \(1 \neq \sqrt{3}\), so the equation does not hold.

### General Analysis:
In general, the equation \(\sqrt{a} - \sqrt{b} = \sqrt{a - b}\) does not hold for arbitrary positive values of \(a\) and \(b\). The correct relationship involves squares:
\[
\sqrt{a} - \sqrt{b} \neq \sqrt{a - b}
\]
for any positive \(a\) and \(b\) where \(a \neq b\).

Therefore, the correct answer is **False**.

Would you like more details or have any questions?

Here are five questions related to this topic:
1. What is the correct interpretation of \(\sqrt{a - b}\) in terms of \(a\) and \(b\)?
2. How would the equation change if we consider the sum of square roots instead?
3. What are some examples where this equation might appear to hold true?
4. Can you derive a condition under which \(\sqrt{a} - \sqrt{b} = \sqrt{a - b}\) might be approximately true?
5. What are the implications of this false equation for solving algebraic problems?

**Tip:** Always check specific cases to verify the validity of an algebraic equation before generalizing.

State whether the equation \(\sqrt{a} - \sqrt{b} = \sqrt{a - b}\) is true or false for \(a, b > 0\).

Explore whether the equation \(\sqrt{a} - \sqrt{b} = \sqrt{a - b}\) holds true for positive values of a and b. This problem delves into key algebraic concepts such as square roots and inequalities, making it suitable for students in Grades 8-10.

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