Let's solve the given mathematical problem step by step:

Problem:

Find the value of $\sqrt{3 - 2\sqrt{2}}$.

Solution:

1. Let $x = \sqrt{3 - 2\sqrt{2}}$.

2. Rewrite $3 - 2\sqrt{2}$ in the form $(\sqrt{a} - \sqrt{b})^2$:

   • Expand $(\sqrt{a} - \sqrt{b})^2 = a + b - 2\sqrt{ab}$.
   • Comparing $3 - 2\sqrt{2}$ with $a + b - 2\sqrt{ab}$, we see:
     • $a + b = 3$,
     • $2\sqrt{ab} = 2\sqrt{2}$, so $\sqrt{ab} = \sqrt{2}$ and $ab = 2$.

3. Solve for $a$ and $b$:

   • $a + b = 3$ and $ab = 2$ imply the quadratic equation: $t^2 - (a+b)t + ab = 0 \implies t^2 - 3t + 2 = 0.$
   • Factorize: $(t - 1)(t - 2) = 0$, so $t = 1$ or $t = 2$.
   • Thus, $a = 2$ and $b = 1$ (or vice versa).

4. Rewrite $3 - 2\sqrt{2}$ as: $\sqrt{3 - 2\sqrt{2}} = \sqrt{(\sqrt{2} - \sqrt{1})^2}.$

5. Simplify: $\sqrt{3 - 2\sqrt{2}} = \sqrt{2} - \sqrt{1}.$

Final Answer:

The correct option is: $\text{(b) } \sqrt{2} - \sqrt{1}.$

Do you have any questions or would you like a more detailed explanation?

Related Questions:

1. How do you rewrite an expression under a square root in the form of $(\sqrt{a} \pm \sqrt{b})^2$?
2. Can you derive the quadratic equation used to find $a$ and $b$?
3. What happens if $3 - 2\sqrt{2}$ cannot be written as a perfect square?
4. How can this method be extended to cube roots?
5. What are other examples of simplifying square root expressions like this one?

Tip:

Always look for patterns and rewrite complex square root terms as perfect squares for simplification!