If `sqrt3=1.732 and sqrt2 =1.414 , " the value of " 1/(sqrt3+sqrt2)` is

Options

A. 0.064
B. 0.308
C. 0.318
D. 2.146

Answer

To solve the problem of finding the value of \( \frac{1}{\sqrt{3} + \sqrt{2}} \), we will rationalize the denominator. Here’s the step-by-step solution:

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

### Step 2: Rationalize the denominator
To rationalize the denominator, we multiply and divide by the conjugate of the denominator, which is \( \sqrt{3} - \sqrt{2} \):
\[ \frac{1}{\sqrt{3} + \sqrt{2}} \cdot \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} \]

### Step 3: Simplify the denominator
Using the difference of squares formula \( (A + B)(A - B) = A^2 - B^2 \), we simplify the denominator:
\[ (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1 \]
Thus, the expression simplifies to:
\[ \sqrt{3} - \sqrt{2} \]

### Step 4: Substitute the values of \( \sqrt{3} \) and \( \sqrt{2} \)
Now we substitute the approximate values given in the problem:
\[ \sqrt{3} = 1.732 \quad \text{and} \quad \sqrt{2} = 1.414 \]
So we have:
\[ \sqrt{3} - \sqrt{2} = 1.732 - 1.414 \]

### Step 5: Perform the subtraction
Now we perform the subtraction:
\[ 1.732 - 1.414 = 0.318 \]

### Final Answer
Thus, the value of \( \frac{1}{\sqrt{3} + \sqrt{2}} \) is:
\[ \boxed{0.318} \]

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