To determine which is smaller, \(\sqrt{2} - 1\) or \(\sqrt{3} - \sqrt{2}\), let's compare them directly.

1. **Square each expression** to avoid dealing with square roots directly.

   For \(\sqrt{2} - 1\):
   \[
   (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}
   \]

   For \(\sqrt{3} - \sqrt{2}\):
   \[
   (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6}
   \]

2. **Compare the squared expressions**:
   \[
   3 - 2\sqrt{2} \quad \text{and} \quad 5 - 2\sqrt{6}
   \]

   Since \( \sqrt{2} \approx 1.414\) and \( \sqrt{6} \approx 2.449\), we can approximate:
   \[
   2\sqrt{2} \approx 2.828 \quad \text{and} \quad 2\sqrt{6} \approx 4.898
   \]
   \[
   3 - 2\sqrt{2} \approx 3 - 2.828 = 0.172
   \]
   \[
   5 - 2\sqrt{6} \approx 5 - 4.898 = 0.102
   \]

3. **Conclusion**: Since \( 0.102 < 0.172 \), \(\sqrt{3} - \sqrt{2}\) is smaller than \(\sqrt{2} - 1\).

So, \(\sqrt{3} - \sqrt{2}\) is smaller.

Would you like further details or have any questions?

Here are 5 related questions you might be interested in:
1. How do you simplify square root expressions involving addition or subtraction?
2. How can you approximate square roots without a calculator?
3. What are other methods to compare expressions involving square roots?
4. How do irrational numbers affect inequalities?
5. Can square roots of non-perfect squares be represented as simple fractions?

**Tip:** When comparing expressions with square roots, sometimes squaring both sides helps simplify the comparison, but be careful to check for extraneous solutions introduced by squaring.

Explore the comparison between √2 - 1 and √3 - √2 to determine which expression is smaller. This mathematical problem involves square roots, inequalities, and approximations. Learn step-by-step how to compare these expressions and understand the underlying concepts. Suitable for advanced high school students (Grades 10-12).

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