If `sqrt(2)=1.4142`… is given, then the value of `(7)/((3+sqrt(2)))` correct upto two decimal places is

Options

A. `1.59`
B. `1.60`
C. `2.58`
D. `2.57`

Answer

To solve the expression \( \frac{7}{3 + \sqrt{2}} \) and find its value up to two decimal places, we can follow these steps:

### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 + \sqrt{2} \) 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 2: Simplify the Denominator
Now we simplify the denominator using the difference of squares formula \( a^2 - b^2 \):

\[ (3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]

### Step 3: Substitute Back into the Expression
Now substitute the simplified denominator back into the expression:

\[ \frac{7(3 - \sqrt{2})}{7} \]

### Step 4: Cancel the Common Factor
The \( 7 \) in the numerator and denominator cancels out:

\[ 3 - \sqrt{2} \]

### Step 5: Substitute the Value of \( \sqrt{2} \)
Now substitute the given value of \( \sqrt{2} = 1.4142 \):

\[ 3 - 1.4142 = 1.5858 \]

### Step 6: Round to Two Decimal Places
Finally, we round \( 1.5858 \) to two decimal places:

\[ 1.59 \]

Thus, the value of \( \frac{7}{3 + \sqrt{2}} \) correct up to two decimal places is **1.59**.