If `sqrt(2) = 1.414`, the square root of `(sqrt(2) - 1)/(sqrt(2) + 1)` is nearest to

Options

A. 0.172
B. 0.414
C. 0.586
D. 1.414

Answer

To solve the problem, we need to find the square root of the expression \((\sqrt{2} - 1)/(\sqrt{2} + 1)\) given that \(\sqrt{2} = 1.414\).

### Step-by-Step Solution:

1. **Substitute the value of \(\sqrt{2}\)**:
\[ \sqrt{2} = 1.414 \]
Substitute this value into the expression:
\[ \frac{\sqrt{2} - 1}{\sqrt{2} + 1} = \frac{1.414 - 1}{1.414 + 1} \]

2. **Calculate the numerator and denominator**:
- Numerator:
\[ 1.414 - 1 = 0.414 \]
- Denominator:
\[ 1.414 + 1 = 2.414 \]

3. **Form the fraction**:
\[ \frac{0.414}{2.414} \]

4. **Calculate the fraction**:
To simplify \(\frac{0.414}{2.414}\), we can perform the division:
\[ \frac{0.414}{2.414} \approx 0.171 \]

5. **Find the square root of the result**:
Now, we need to find the square root of \(0.171\):
\[ \sqrt{0.171} \approx 0.414 \]

6. **Determine the nearest option**:
From the given options (0.172, 0.414, 0.586, and 1.414), the value \(0.414\) is the closest to our calculated result.

### Final Answer:
The square root of \(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\) is nearest to **0.414**.