if `sqrt(2)=1.4142` the value of `7/(3+sqrt2)`

Options

A. 1.5858
B. 4.4142
C. 3.4852
D. 3.5858

Answer

To solve the problem \( \frac{7}{3 + \sqrt{2}} \) given that \( \sqrt{2} = 1.4142 \), we will rationalize the denominator. Here’s a step-by-step solution:

### Step 1: Write the expression
We start with the expression:
\[ \frac{7}{3 + \sqrt{2}} \]

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 3 - \sqrt{2} \):
\[ \frac{7}{3 + \sqrt{2}} \cdot \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{7(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \), we calculate the denominator:
\[ (3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ 7(3 - \sqrt{2}) = 21 - 7\sqrt{2} \]

### Step 5: Combine the results
Putting it all together, we have:
\[ \frac{21 - 7\sqrt{2}}{7} \]

### Step 6: Separate the terms
We can separate the terms in the fraction:
\[ \frac{21}{7} - \frac{7\sqrt{2}}{7} = 3 - \sqrt{2} \]

### Step 7: Substitute the value of \( \sqrt{2} \)
Now we substitute \( \sqrt{2} = 1.4142 \):
\[ 3 - 1.4142 = 1.5858 \]

### Final Answer
Thus, the value of \( \frac{7}{3 + \sqrt{2}} \) is approximately:
\[ \boxed{1.5858} \]