The value of `sqrt((3-2sqrt(2)))` is

Options

A. `sqrt(3) + sqrt(2)`
B. `1-sqrt(2)`
C. `sqrt(2) + 1`
D. `sqrt(2)-1`

Answer

To find the value of \( \sqrt{3 - 2\sqrt{2}} \), we can simplify the expression step by step.

### Step-by-Step Solution:

1. **Rewrite the expression**:
We start with the expression \( \sqrt{3 - 2\sqrt{2}} \).

2. **Identify a perfect square**:
We can express \( 3 \) as \( 2 + 1 \):
\[ 3 - 2\sqrt{2} = (2 + 1) - 2\sqrt{2} \]

3. **Recognize the form of a square**:
Notice that \( 2\sqrt{2} \) can be rewritten as \( 2 \cdot 1 \cdot \sqrt{2} \). This suggests that we can express the entire expression as a square:
\[ 3 - 2\sqrt{2} = (\sqrt{2} - 1)^2 \]
Here, \( a = \sqrt{2} \) and \( b = 1 \). Thus, we can confirm:
\[ a^2 + b^2 - 2ab = (\sqrt{2})^2 + (1)^2 - 2(\sqrt{2})(1) = 2 + 1 - 2\sqrt{2} = 3 - 2\sqrt{2} \]

4. **Take the square root**:
Now we can take the square root of both sides:
\[ \sqrt{3 - 2\sqrt{2}} = \sqrt{(\sqrt{2} - 1)^2} \]

5. **Simplify the square root**:
The square root of a square gives us the absolute value:
\[ \sqrt{3 - 2\sqrt{2}} = |\sqrt{2} - 1| \]
Since \( \sqrt{2} \) is approximately \( 1.414 \), which is greater than \( 1 \), we have:
\[ |\sqrt{2} - 1| = \sqrt{2} - 1 \]

### Final Answer:
Thus, the value of \( \sqrt{3 - 2\sqrt{2}} \) is:
\[ \sqrt{2} - 1 \]